Numerical Performance of Triangle Element Approximation for Solving 2D Poisson Equations Using 4-Point EDGAOR Method

Size: px
Start display at page:

Download "Numerical Performance of Triangle Element Approximation for Solving 2D Poisson Equations Using 4-Point EDGAOR Method"

Transcription

1 International Journal of Mathematical Analysis Vol. 9, 2015, no. 54, HIKARI Ltd, Numerical Performance of Triangle Element Approximation for Solving 2D Poisson Equations Using 4-Point EDGAOR Method Mohd Kamalrulzaman Md Akhir, Jumat Sulaiman Faculty of Science and Natural Resources Universiti Malaysia Sabah Kota Kinabalu Sabah Malaysia Copyright c 2015 Mohd Kamalrulzaman Md Akhir and Jumat Sulaiman. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract This paper aims to examine the implementation of the Explicit Decoupled Group (EDG) methods. More specifically, the four-point Explicit Decoupled Group Accelerated Over-relaxation (4-EDGAOR) is investigated. Owing to the efficacy of this method, the main aim of the current research is to demonstrate the benefits of the 4-EDGAOR in solving two-dimensional (2D) Poisson equations with the help of the half-sweep triangle finite element approximation equation based on the Galerkin scheme. The results of numerical experiments demonstrate the effectiveness of the 4-EDGAOR method over the previous four point block methods (4-EDGSOR, 4-EDG, 4-EGAOR, 4-EGSOR and 4-EGGS). Based on the numerical results obtained, the results show that the 4- EDGAOR method outperforms the 4-EDGSOR, 4-EDG, 4-EGAOR, 4- EGSOR and 4-EG methods in terms of number of iterations and CPU time. Mathematics Subject Classification: 41A55, 45A05, 45B05 Keywords: Partial Differential equations (PDEs); Poisson equation, Explicit Decoupled Group (EDG), Point Block Iteration; Galerkin Scheme, Triangle element, AOR method

2 2668 Mohd Kamalrulzaman Md Akhir and Jumat Sulaiman 1 Introduction Partial Differential equations (PDEs) are implemented in mathematical models in numerous and diverse physical circumstances, as well as in re-formulations of other mathematical problems. As evident in the related literature, PDEs of the Poisson type are among the most practical and frequently investigated. Therefore, in this paper, numerical solutions of linear, 2D Poisson equations are considered. The standard form for 2D Poisson equations can be represented mathematically as follows: 2 U x 2 with the dirichlet boundary conditions + 2 U = f (x, y), (x, y) a, b a, b (1) y2 U(x, a) = g 1 (x), a x b, U(x, b) = g 2 (x), a x b, U(a, y) = g 3 (y), a x b, U(b, y) = g 4 (x), a y b, where f (x, y) is a given function with sufficient smoothness. A numerical approach to the solution of the problem (1) is a fundamental subdivision of scientific examination. Basically, 2D Poisson equations are solved numerically by discretizing the problems to the solution of linear systems. In order to obtain numerical solutions, some valid numerical methods for discretizing the problem (1) have been developed in recent years, including the mesh-free 4, 20, 21 and mesh-based methods. However, these discretization schemes mostly lead to large sparse linear systems that could involve high price in order to provide the solution, based on direct methods, with the escalation in the order of the linear systems. Hence, a substitute of these schemes could be the iterative methods that provide proficient solutions for the above-mentioned extensive problems Amongst the existing iterative methods, point block methods have been extensively accepted as efficient methods for sparse linear systems. The standard block method (also known as the 4-EGGS method 6) is a particular example of the four point block method. Apart from the standard 4-EGGS method, the variants of the four point block method, which are 4-EDGSOR methods, have also been proposed by 1. Additional research studies were executed by 9, 10, 12, to authenticate the efficacy of the half-sweep concept. Fundamentally, the EDG method is derived from a complexity reduction approach based on half- sweep concepts, respectively. To introduce the 4-EDGAOR method based on the Galerkin scheme in solving 2D Poisson equations, is the foremost objective of current research paper. As shown in Figure 1, in order to simplify the formulation of the full-sweep and half-sweep triangle element approximation equations for problem (1), uni-

3 Numerical performance of triangle element approximation 2669 (a) (b) Figure 1: (a) and (b) show the solution domain Ω of triangle elements for the full- and half-sweep cases at n = 8. form node points are discussed only. Based on Figure 1, there is a need to discretize the solution of domain evenly in both the x and y directions with a mesh size h, as given below: x = y = h = b a, m = n + 1. (2) n As depicted in Figure 1, for problem (1), the triangle finite element networks were built as a guide to derive triangle finite element approximation equations. Correspondingly, the similar concept of the half-sweep iterations was implemented in the finite difference methods 1, 16; each triangle element involved only two node points of type, as revealed in Figure 1. Thus, the full-sweep and half-sweep approaches were implemented in the same type of node points until the iterative convergence test was achieved. Afterwards, at the remaining points (points of type ), other approximate solutions were computed directly 1, 13, 16. The framework of this paper is ordered in the sequence explained subsequently. Starting with Section 2, derivation of the half-sweep triangle element approximation is elaborated at first. Followed by the Section 3 that discusses the implementation of the 4-EDGAOR method for solving problems (1). Subsequently, in Section 4, numerical results are shown to assess the performance of the examined iterative methods followed by the concluding remarks that are presented in Section 5.

4 2670 Mohd Kamalrulzaman Md Akhir and Jumat Sulaiman 2 Formulation of Half-sweep Triangle Element Approximations As discussed in the aforementioned section, to solve 2D Poisson equations, the 4-EDGAOR method was implemented by using the half-sweep finite element approximation equation based on the Galerkin scheme. The common approximation of the function U(x, y), by keeping in view only three node points of type, in the form of an interpolation function for a haphazard triangle element, e, is given by 3, 4: Ũ e (x, y) = N 1 (x, y)u 1 + N 2 (x, y)u 2 + N 3 (x, y)u 3 (3) The shape functions N k (x, y), k = 1, 2, 3, can generally be shown as: where, N k (x, y) = 1 A (a k + b k x + c k y), k = 1, 2, 3 (4) A = x 1 (y 2 y 3 ) + x 2 (y 3 y 1 ) + x 3 (y 1 y 2 ) a 1 a 2 a 3 = x 2 y 3 x 3 y 2 x 3 y 1 x 1 y 3 x 1 y 2 x 2 y 1, b 1 b 2 b 3 = y 2 y 3 y 3 y 1 y 1 y 2, c 1 c 2 c 3 = x 3 x 2 x 1 x 3 x 2 x 3 Beside this, the rst order partial derivatives of the shape functions towards x and y can be shown, respectively, as follows: (N x k (x, y)) = y (N k (x, y)) = b k det c A k det A, }, k = 1, 2, 3 (5) According to Figure 2, using the approximation of the functions U(x, y) and f(x, y), the distribution of the hat function, R r,s (x, y), in the solution domain, in the case of the full- and half-sweep for the whole domain can be defined as given below and was also described in 19: Ũ(x, y) = f(x, y) = m m R r,s (x, y)u r,s (6) r=0 s=0 m r=0 s=0 m R r,s (x, y)u r,s (7)

5 Numerical performance of triangle element approximation 2671 (a) (b) Figure 2: (a) and (b) show the definition of the hat function R i,j (x, y), of fulland half-sweep triangle elements at the solution domain. and Ũ(x, y) = f(x, y) = m m r=0,2,4 s=0,2,4 m m r=0,2,4 s=0,2,4 R r,s (x, y)u r,s + R r,s (x, y)u r,s + m 1 m 1 r=1,3,5 s=1,3,5 m 1 m 1 r=1,3,5 s=1,3,5 R r,s (x, y)u r,s (8) R r,s (x, y)u r,s (9) Hence, Eq. (8) is is an approximate solution of the problem (1). For problem (1), in order to develop the full-sweep and half-sweep finite element approximation equations, Galerkin scheme 18 was considered as follows: R i,j (x, y)e i,j (x, y) = 0, i, j = 0, 1, 2,..., m (10) D + 2 U y 2 where, E(x, y) = 2 U x 2 the Green theorem, Eq. (10) can be rewritten as: λ ( R i,j (x, y) u f(x, y) is a residual function. By incorporating y dx + R i,j(x, y) u ) x dy b b ( Ri,j (x, y) u x x + R i,j(x, y) y a a ) u dxdy = F i,j (11) y Then by simplying Eq. (10), we can derive the finite element approximation equation gives as follows: K i,j,r,su r,s = C i,j,r,sf r,s (12)

6 2672 Mohd Kamalrulzaman Md Akhir and Jumat Sulaiman where, K i,j,r,s = C i,j,r,s = b b ( Ri,j a a b b a a x ) R r,s dxdy + x ( Ri,j (x, y)r r,s (x, y) ) dxdy. b b ( Ri,j a a y ) R r,s dxdy, y As a matter of fact, for the full- and half-sweep cases, the linear system in Eq. (12) could be simply articulated in the form of stencil, as given below in Eq. (13) and Eq. (14): Full-sweep: Half-sweep: U i,j = h f i,j (13) U i,j = h2 5 1 f i,j, i = U i,j = h f i,j, i 1, n U i,j = h2 1 5 f i,j, i = n (14) 6 In fact, the stencil forms in Eqs. (13) and (14) forms consist of seven node points in formulating their approximation equations. On the other hand, two of its coefficients are zero. Apart of this, the stencil forms for both triangle finite element schemes are the same compared to the existing five points finite difference scheme, see 1, The AOR Method The following discussion can be found in 2, 5, Formulation of 4-EGAOR Method The following discussion can be found in 14.

7 Numerical performance of triangle element approximation Formulation of 4-EDGAOR Method According to Abdullah 1, in solving the 2D Poisson equation via the halfsweep finite element approximation equation, the 4-EDG method proved to be more efficient as compared to the 4-EG method. Likewise, the same steps were adopted for the finite difference approach. Let a four solid point group be selected to develop a (4x4) linear system, as shown below: where, U i,j U i+1,j+1 U i+1,j U i,j+1 = S 1 S 2 S 3 S 4 (15) S 1 = U i 1,j 1 + U i 1,j+1 + U i+1,j 1 F i,j+1, S 2 = U i+2,j + U i,j+2 + U i+2,j+2 F i+1,j+1, S 3 = U i,j 1 + U i+2,j 2 + U i+2,j+1 F i+1,j, S 4 = U i 1,j + U i 1,j+2 + U i+1,j+2 F i,j+1, and, F i,j = h2 6 (f i 2,j + f i+2,j + f i 1,j 1 + +f i 1,j+1 + f i+1,j 1 + f i+1,j+1 + 6f i,j ) The linear system in Eq. (15) can be independently decomposed into two (2x2) linear systems. Therefore, the 4-EDG method can be easily reduced as follows: u i,j u i+1,j+1 ui+1,j u i,j+1 (k+1) = (k+1) = S1 S 2 S3 S 4 (16) (17) By adding the parameter, ω into Eqs. (16) and (17), the 4-EDGSOR method can be simplified to become: u i,j u i+1,j+1 (k+1) = w S1 S 2 + (1 ω) u i,j u i+1,j+1 (k) (18) ui+1,j u i,j+1 (k+1) = w S3 S 4 ui+1,j + (1 ω) u i,j+1 (k) (19)

8 2674 Mohd Kamalrulzaman Md Akhir and Jumat Sulaiman where the value is defined in the range, of 1 ω < 2. Now, we investigate the performance of the 4-EDGAOR method which is derived on the combination between the 4-EDG and AOR methods. Therefore, by applying the AOR method 2, 12, 21 into Eqs. (18) and (19), the general scheme for this method can be shown as follows: U i,j U i+1,j+1 (k+1) = r 4T1 15 T 1 + ω S1 S 2 + (1 ω) U i,j U i+1,j+1 (k) (20) Ui+1,j where, U i,j+1 (k+1) = r 4T1 15 T 1 + ω S3 S 4 Ui+1,j + (1 ω) U i,j+1 (k) (21) T 1 = U (k+1) i 1,j 1 U (k) i 1,j 1 + U (k+1) i+1,j 1 U (k) i+1,j 1 + U (k+1) i 1,j+1 U (k) i 1,j+1, S 1 = U (k) i 1,j 1 + U (k) i 1,j+1 + U (k) i+1,j 1 F i,j+1, S 2 = U (k) i+2,j + U (k) i,j+2 + U (k) i+2,j+2 F i+1,j+1, S 3 = U (k) i,j 1 + U (k) i+2,j 2 + U (k) i+2,j+1 F i+1,j, S 4 = U (k) i 1,j + U (k) i 1,j+2 + U (k) i+1,j+2 F i,j+1. Because of the extra benefits of the AOR method, which have two weighted parameters, all of the common existing methods become unique cases of this method in the scenario the parameters take certain values. For example, when w = 1 and r = 0, we acquire the the point block Jacobi method. If w = r = 1, we acquire the point block GS method. If w = r, the point block SOR method is attained 21. Since, the coefficient matrix in Eq. (12) is a pentadiagonal matrix, it has the property A, and is Consistently Ordered 7. At this moment, to implement 4-EDGAOR method, we use Eq. (3.2) or Eq. (3.2) allows us to iterate through half of the points, lying on the 2h-grid. Again, it can be observed that Eq. (3.2) or (3.2) involves a group of points of type. To implement the iteration process, the algorithmn of the 4-EDGAOR method can be displayed as follows: 1. Discretize the solution domain into point of types (ie., ) as shown in Figure 1(b).

9 Numerical performance of triangle element approximation Perform iterations (using Eqs. (3.2) or (3.2)), taking the values of r = ω from the segment 1, 2). 3. Within the interval 0.1 from the value found in Step 2, define the optimal ω opt with a precision of 0.01 by choosing consecutive values for which k is minimal; r is taken to be equal to ω. 4. Perform experiments using the value of ω opt and choosing consecutive values of r with a precision of 0.01 within the interval 0.1 from the ω opt. 5. Define the value r opt for which k is minimal. 6. Evaluate the solutions at the remaining point of type using Eq. (13). 7. Display approximate solutions. 4 Numerical Results In order to compare the recitals of the methods described in the previous sections, several experiments were carried out on the following 2D Poisson example 11: 2 U x U y 2 = (cos(x + y) + cos(x y)). (22) and the exact solution is given by U(x, 0) = cos x, U ( x, π 2 ) = 0, U(0, y) = cos y, U(π, y) = cos y. U(x, y) = cos(x) cos(y). The numerical experiments were carried out on a dedicated personal with an PC Intel(R) Core (TM) i7 CPU 860@3.00Ghz, and 6.00GB RAM. The programming codes were written in C++ programming language. The AOR method obtained in this paper is compared to other methods (4-EDGSOR, 4-EDG, 4-EGAOR, 4-EGSOR and 4-EG). The value of the initial iteration is set to be zero for the test problems and in the course of implementation the tolerance error, is considered ε = For convenience, there are three vital parameters to be measured, including the number of iterations (k), maximum absolute error (Abs.Error) and the execution time (t in seconds). The numerical results of the experiment for the proposed iterative methods are given in Table 1.

10 2676 Mohd Kamalrulzaman Md Akhir and Jumat Sulaiman Table 1: Comparison of the number of iterations, execution time (seconds) and maximum absolute error for the iterative methods. n Methods r w k t Abs.Error 4-EG e-7 4-EDG e-6 4-EGSOR e EDGSOR e-6 4-EGAOR e-7 4-EDGAOR e-6 4-EG e-7 4-EDG e-6 4-EGSOR e EDGSOR e-6 4-EGAOR e-7 4-EDGAOR e-6 4-EG e-7 4-EDG e-6 4-EGSOR e EDGSOR e-6 4-EGAOR e-7 4-EDGAOR e-6 4-EG e-7 4-EDG e-6 4-EGSOR e EDGSOR e-6 4-EGAOR e-7 4-EDGAOR e-6

11 Numerical performance of triangle element approximation Conclusions In this paper, we have presented an application of the 4-EDGAOR method for solving sparse linear systems generated from the discretization of the 2D Poisson equation equations by using the Galerkin scheme. The numerical results obtained for the proposed problem (Table 1) clearly show that applying the AOR methods reduces the number of iterations, and execution time, compared to the SOR and GS methods. At the same time, it has been shown that applying the half -sweep approach reduces the computational time in the implementation of the iterative method. Overall, the numerical results demonstrate that the 4-EDGAOR method outperforms the existing block methods (4-EDGSOR, 4-EDG, 4-EGAOR, 4- EGSOR and 4-EG), particularly in the sense of the number of iterations and execution time. This is mainly attributable to the reduction of the computational complexity; since the implementations of the 4-EDGAOR method only consider approximately half of all interior node points in a solution domain. For future work, the capability of the quarter-sweep approach 2, 12, 21 should be investigated in terms of the point block iterative method by using the Galerkin scheme. References 1 A.R. Abdullah, The Four Point Explicit Decoupled Group (EDG) Method: A Fast Poisson Solver, Intern. J. of Comp. Math., 38 (1991), A. Hadjidimos, Accelerated Over Relaxation method, Math. of Comput., 32 (1978), C.A.J. Fletcher, The Galerkin method: An introduction. In: Noye, J. (pnyt.), Num. Simul. of Flu. Mot., Amsterdam, North-Holland Publishing Company, 52 (1978), C.A.J. Fletcher, Computational Galerkin Method. Springer Series in Computational Physics, Springer-Verlag, New York, D. J. Evans and M.M. Martins, The AOR method For AX - XB = C, Int. J. of Comp. Math., 52 (1994), D. J. Evans, Group Explicit Methods for the Numerical Solutions of Partial Differential Equations, Gordon and Breach Science Publishers, Australia, 1997.

12 2678 Mohd Kamalrulzaman Md Akhir and Jumat Sulaiman 7 D.M. Young, Iterative Solution of Large Linear Systems, Academic Press, London, G. Yagawa and T. Furukawa, Recent developments of free mesh method, Int. J. for Numer. Meth. in Engi., 47 (2000), J. Sulaiman, M.K. Hasan and M. Othman, Red-Black Half-Sweep Iterative Method Using Triangle Finite Element Approximation for 2D Poisson Equations, Lecture Notes in Computer Science, 4487 (2007), J. Sulaiman, M.K. Hasan, M. Othman, Red-Black EDGSOR Iterative Method Using Triangle Element Approximation for 2D Poisson Equations, Lecture Notes in Computer Science, 4707 (2007), M.K.M. Akhir, J. Sulaiman, Triangle Element Analysis for the Solutions of 2D Poisson Equations via AOR method, J. of Adv. in Math., 11 (2015), no. 2, M.K.M. Akhir, J. Sulaiman, HSAOR Iterative Method for the Finite Element Solution of 2D Poisson Equations, Int. J. of Math. And Comp., 27 (2016), no. 2, M.K.M. Akhir., J. Sulaiman, The 4-EGAOR Method for Solving Triangle Element Approximation of 2D Poisson Equations, App. Math. Scie., 9 (2015), M.K.M. Akhir, J. Sulaiman, Analysis of Triangle Element Approximation fopr Solving 2D Poisson Equations using QSAOR, Global Journal of Pure and App. Math., 2015, In Press. 15 O.C. Zienkiewicz, Why finite elements?, Finite Elements in Fluids Volume, ln. R.H. Gallagher, J.T. Oden, C.Taylor, O.C. Zienkiewicz, (Eds.), John Wiley and Sons, London, P.E. Lewis and J.P. Ward, The Finite Element Method: Principles and Applications, Addison-Wesley Publishing Company, Wokingham, R. Vichnevetsky, Computer Methods for Partial Differential Equations, Vol I, Prentice-Hall, New Jersey, 1981.

13 Numerical performance of triangle element approximation T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, and P. Krysl, Meshless methods: An overview and recent developments, Comp. Meth. in App. Mech. and Eng., 139 (1996), T. Zhu, A New Meshless Regular Local Boundary Integral Equation (MRLBIE) approach, Int. Journal for Num. Meth.s in Eng., 46 (1999), W.S. Yousif and D.J. Evans, Explicit Decoupled Group Iterative Methods and Their Implementations, Parallel Algorithms and Applications, 7 (1995), W.S. Yousif and M.M. Martins, Explicit De-couple Group AOR method for solving elliptic partial differential equations, Neural, Parallel and Scientific Computations, 16 (2008), no. 4, Received: October 4, 2015; Published: December 2, 2015

Path Planning for Indoor Mobile Robot using Half-Sweep SOR via Nine-Point Laplacian (HSSOR9L)

Path Planning for Indoor Mobile Robot using Half-Sweep SOR via Nine-Point Laplacian (HSSOR9L) IOSR Journal of Mathematics (IOSR-JM) ISSN: 2278-5728. Volume 3, Issue 2 (Sep-Oct. 2012), PP 01-07 Path Planning for Indoor Mobile Robot using Half-Sweep SOR via Nine-Point Laplacian (HSSOR9L) Azali Saudi

More information

Explicit Decoupled Group Iterative Method for the Triangle Element Solution of 2D Helmholtz Equations

Explicit Decoupled Group Iterative Method for the Triangle Element Solution of 2D Helmholtz Equations Interntionl Mthemticl Forum, Vol. 12, 2017, no. 16, 771-779 HIKARI Ltd, www.m-hikri.com https://doi.org/10.12988/imf.2017.7654 Explicit Decoupled Group Itertive Method for the Tringle Element Solution

More information

Cubic B-spline Solution of Two-point Boundary Value Problem Using HSKSOR Iteration

Cubic B-spline Solution of Two-point Boundary Value Problem Using HSKSOR Iteration Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 11 (2017), pp. 7921-7934 Research India Publications http://www.ripublication.com Cubic B-spline Solution of Two-point Boundary

More information

The Number of Fuzzy Subgroups of Cuboid Group

The Number of Fuzzy Subgroups of Cuboid Group International Journal of Algebra, Vol. 9, 2015, no. 12, 521-526 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2015.5958 The Number of Fuzzy Subgroups of Cuboid Group Raden Sulaiman Department

More information

Application of Four-Point MEGMSOR Method for. the Solution of 2D Helmholtz Equations

Application of Four-Point MEGMSOR Method for. the Solution of 2D Helmholtz Equations International Journal of Mathematical Analysis Vol. 9, 05, no. 38, 847-86 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.988/ijma.05.5670 Application of Four-Point MEGMOR Method for the olution of D

More information

Solving a Two Dimensional Unsteady-State. Flow Problem by Meshless Method

Solving a Two Dimensional Unsteady-State. Flow Problem by Meshless Method Applied Mathematical Sciences, Vol. 7, 203, no. 49, 242-2428 HIKARI Ltd, www.m-hikari.com Solving a Two Dimensional Unsteady-State Flow Problem by Meshless Method A. Koomsubsiri * and D. Sukawat Department

More information

3D Nearest-Nodes Finite Element Method for Solid Continuum Analysis

3D Nearest-Nodes Finite Element Method for Solid Continuum Analysis Adv. Theor. Appl. Mech., Vol. 1, 2008, no. 3, 131-139 3D Nearest-Nodes Finite Element Method for Solid Continuum Analysis Yunhua Luo Department of Mechanical & Manufacturing Engineering, University of

More information

Robot Path Planning with EGSOR Iterative Method using Laplacian Behaviour-Based Control (LBBC)

Robot Path Planning with EGSOR Iterative Method using Laplacian Behaviour-Based Control (LBBC) 2014 Fifth International Conference on Intelligent Systems, Modelling and Simulation Robot Path Planning with EGSOR Iterative Method using Laplacian Behaviour-Based Control (LBBC) Azali Saudi School of

More information

f xx + f yy = F (x, y)

f xx + f yy = F (x, y) Application of the 2D finite element method to Laplace (Poisson) equation; f xx + f yy = F (x, y) M. R. Hadizadeh Computer Club, Department of Physics and Astronomy, Ohio University 4 Nov. 2013 Domain

More information

AMS527: Numerical Analysis II

AMS527: Numerical Analysis II AMS527: Numerical Analysis II A Brief Overview of Finite Element Methods Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao SUNY Stony Brook AMS527: Numerical Analysis II 1 / 25 Overview Basic concepts Mathematical

More information

Graceful Labeling for Some Star Related Graphs

Graceful Labeling for Some Star Related Graphs International Mathematical Forum, Vol. 9, 2014, no. 26, 1289-1293 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.4477 Graceful Labeling for Some Star Related Graphs V. J. Kaneria, M.

More information

Hyperbola for Curvilinear Interpolation

Hyperbola for Curvilinear Interpolation Applied Mathematical Sciences, Vol. 7, 2013, no. 30, 1477-1481 HIKARI Ltd, www.m-hikari.com Hyperbola for Curvilinear Interpolation G. L. Silver 868 Kristi Lane Los Alamos, NM 87544, USA gsilver@aol.com

More information

Fully discrete Finite Element Approximations of Semilinear Parabolic Equations in a Nonconvex Polygon

Fully discrete Finite Element Approximations of Semilinear Parabolic Equations in a Nonconvex Polygon Fully discrete Finite Element Approximations of Semilinear Parabolic Equations in a Nonconvex Polygon Tamal Pramanick 1,a) 1 Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati

More information

Building the Graphics Memory of. the Stiffness Matrix of the Beam

Building the Graphics Memory of. the Stiffness Matrix of the Beam Contemporary Engineering Sciences, Vol. 11, 2018, no. 92, 4593-4605 HIKARI td, www.m-hikari.com https://doi.org/10.12988/ces.2018.89502 Building the Graphics Memory of the Stiffness Matrix of the Beam

More information

Parallel Implementations of Gaussian Elimination

Parallel Implementations of Gaussian Elimination s of Western Michigan University vasilije.perovic@wmich.edu January 27, 2012 CS 6260: in Parallel Linear systems of equations General form of a linear system of equations is given by a 11 x 1 + + a 1n

More information

Sequences of Finite Vertices of Fuzzy Topographic Topological Mapping

Sequences of Finite Vertices of Fuzzy Topographic Topological Mapping Applied Mathematical Sciences, Vol. 10, 2016, no. 38, 1923-1934 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2016.6126 Sequences of Finite Vertices of Fuzzy Topographic Topological Mapping

More information

What is the Optimal Bin Size of a Histogram: An Informal Description

What is the Optimal Bin Size of a Histogram: An Informal Description International Mathematical Forum, Vol 12, 2017, no 15, 731-736 HIKARI Ltd, wwwm-hikaricom https://doiorg/1012988/imf20177757 What is the Optimal Bin Size of a Histogram: An Informal Description Afshin

More information

An Investigation into Iterative Methods for Solving Elliptic PDE s Andrew M Brown Computer Science/Maths Session (2000/2001)

An Investigation into Iterative Methods for Solving Elliptic PDE s Andrew M Brown Computer Science/Maths Session (2000/2001) An Investigation into Iterative Methods for Solving Elliptic PDE s Andrew M Brown Computer Science/Maths Session (000/001) Summary The objectives of this project were as follows: 1) Investigate iterative

More information

Performance Comparison between Blocking and Non-Blocking Communications for a Three-Dimensional Poisson Problem

Performance Comparison between Blocking and Non-Blocking Communications for a Three-Dimensional Poisson Problem Performance Comparison between Blocking and Non-Blocking Communications for a Three-Dimensional Poisson Problem Guan Wang and Matthias K. Gobbert Department of Mathematics and Statistics, University of

More information

Wavelet-Galerkin Solutions of One and Two Dimensional Partial Differential Equations

Wavelet-Galerkin Solutions of One and Two Dimensional Partial Differential Equations VOL 3, NO0 Oct, 202 ISSN 2079-8407 2009-202 CIS Journal All rights reserved http://wwwcisjournalorg Wavelet-Galerkin Solutions of One and Two Dimensional Partial Differential Equations Sabina, 2 Vinod

More information

Module 1: Introduction to Finite Difference Method and Fundamentals of CFD Lecture 13: The Lecture deals with:

Module 1: Introduction to Finite Difference Method and Fundamentals of CFD Lecture 13: The Lecture deals with: The Lecture deals with: Some more Suggestions for Improvement of Discretization Schemes Some Non-Trivial Problems with Discretized Equations file:///d /chitra/nptel_phase2/mechanical/cfd/lecture13/13_1.htm[6/20/2012

More information

course outline basic principles of numerical analysis, intro FEM

course outline basic principles of numerical analysis, intro FEM idealization, equilibrium, solutions, interpretation of results types of numerical engineering problems continuous vs discrete systems direct stiffness approach differential & variational formulation introduction

More information

An Optimization Method Based On B-spline Shape Functions & the Knot Insertion Algorithm

An Optimization Method Based On B-spline Shape Functions & the Knot Insertion Algorithm An Optimization Method Based On B-spline Shape Functions & the Knot Insertion Algorithm P.A. Sherar, C.P. Thompson, B. Xu, B. Zhong Abstract A new method is presented to deal with shape optimization problems.

More information

Introduction to Multigrid and its Parallelization

Introduction to Multigrid and its Parallelization Introduction to Multigrid and its Parallelization! Thomas D. Economon Lecture 14a May 28, 2014 Announcements 2 HW 1 & 2 have been returned. Any questions? Final projects are due June 11, 5 pm. If you are

More information

An introduction to mesh generation Part IV : elliptic meshing

An introduction to mesh generation Part IV : elliptic meshing Elliptic An introduction to mesh generation Part IV : elliptic meshing Department of Civil Engineering, Université catholique de Louvain, Belgium Elliptic Curvilinear Meshes Basic concept A curvilinear

More information

An explicit feature control approach in structural topology optimization

An explicit feature control approach in structural topology optimization th World Congress on Structural and Multidisciplinary Optimisation 07 th -2 th, June 205, Sydney Australia An explicit feature control approach in structural topology optimization Weisheng Zhang, Xu Guo

More information

Deficient Quartic Spline Interpolation

Deficient Quartic Spline Interpolation International Journal of Computational Science and Mathematics. ISSN 0974-3189 Volume 3, Number 2 (2011), pp. 227-236 International Research Publication House http://www.irphouse.com Deficient Quartic

More information

1 2 (3 + x 3) x 2 = 1 3 (3 + x 1 2x 3 ) 1. 3 ( 1 x 2) (3 + x(0) 3 ) = 1 2 (3 + 0) = 3. 2 (3 + x(0) 1 2x (0) ( ) = 1 ( 1 x(0) 2 ) = 1 3 ) = 1 3

1 2 (3 + x 3) x 2 = 1 3 (3 + x 1 2x 3 ) 1. 3 ( 1 x 2) (3 + x(0) 3 ) = 1 2 (3 + 0) = 3. 2 (3 + x(0) 1 2x (0) ( ) = 1 ( 1 x(0) 2 ) = 1 3 ) = 1 3 6 Iterative Solvers Lab Objective: Many real-world problems of the form Ax = b have tens of thousands of parameters Solving such systems with Gaussian elimination or matrix factorizations could require

More information

Surrogate Gradient Algorithm for Lagrangian Relaxation 1,2

Surrogate Gradient Algorithm for Lagrangian Relaxation 1,2 Surrogate Gradient Algorithm for Lagrangian Relaxation 1,2 X. Zhao 3, P. B. Luh 4, and J. Wang 5 Communicated by W.B. Gong and D. D. Yao 1 This paper is dedicated to Professor Yu-Chi Ho for his 65th birthday.

More information

Some Algebraic (n, n)-secret Image Sharing Schemes

Some Algebraic (n, n)-secret Image Sharing Schemes Applied Mathematical Sciences, Vol. 11, 2017, no. 56, 2807-2815 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.710309 Some Algebraic (n, n)-secret Image Sharing Schemes Selda Çalkavur Mathematics

More information

Meshless physical simulation of semiconductor devices using a wavelet-based nodes generator

Meshless physical simulation of semiconductor devices using a wavelet-based nodes generator Meshless physical simulation of semiconductor devices using a wavelet-based nodes generator Rashid Mirzavand 1, Abdolali Abdipour 1a), Gholamreza Moradi 1, and Masoud Movahhedi 2 1 Microwave/mm-wave &

More information

A Cumulative Averaging Method for Piecewise Polynomial Approximation to Discrete Data

A Cumulative Averaging Method for Piecewise Polynomial Approximation to Discrete Data Applied Mathematical Sciences, Vol. 1, 16, no. 7, 331-343 HIKARI Ltd, www.m-hiari.com http://dx.doi.org/1.1988/ams.16.5177 A Cumulative Averaging Method for Piecewise Polynomial Approximation to Discrete

More information

Conditional Volatility Estimation by. Conditional Quantile Autoregression

Conditional Volatility Estimation by. Conditional Quantile Autoregression International Journal of Mathematical Analysis Vol. 8, 2014, no. 41, 2033-2046 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.47210 Conditional Volatility Estimation by Conditional Quantile

More information

A meshfree weak-strong form method

A meshfree weak-strong form method A meshfree weak-strong form method G. R. & Y. T. GU' 'centre for Advanced Computations in Engineering Science (ACES) Dept. of Mechanical Engineering, National University of Singapore 2~~~ Fellow, Singapore-MIT

More information

Stochastic Coalitional Games with Constant Matrix of Transition Probabilities

Stochastic Coalitional Games with Constant Matrix of Transition Probabilities Applied Mathematical Sciences, Vol. 8, 2014, no. 170, 8459-8465 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.410891 Stochastic Coalitional Games with Constant Matrix of Transition Probabilities

More information

Index. C m (Ω), 141 L 2 (Ω) space, 143 p-th order, 17

Index. C m (Ω), 141 L 2 (Ω) space, 143 p-th order, 17 Bibliography [1] J. Adams, P. Swarztrauber, and R. Sweet. Fishpack: Efficient Fortran subprograms for the solution of separable elliptic partial differential equations. http://www.netlib.org/fishpack/.

More information

Domination Number of Jump Graph

Domination Number of Jump Graph International Mathematical Forum, Vol. 8, 013, no. 16, 753-758 HIKARI Ltd, www.m-hikari.com Domination Number of Jump Graph Y. B. Maralabhavi Department of Mathematics Bangalore University Bangalore-560001,

More information

Nodal Integration Technique in Meshless Method

Nodal Integration Technique in Meshless Method IOSR Journal of Mechanical and Civil Engineering (IOSR-JMCE) e-issn: 2278-1684,p-ISSN: 2320-334X, Volume 11, Issue 1 Ver. IV (Feb. 2014), PP 18-26 Nodal Integration Technique in Meshless Method Ahmed MJIDILA

More information

MESHLESS SOLUTION OF INCOMPRESSIBLE FLOW OVER BACKWARD-FACING STEP

MESHLESS SOLUTION OF INCOMPRESSIBLE FLOW OVER BACKWARD-FACING STEP Vol. 12, Issue 1/2016, 63-68 DOI: 10.1515/cee-2016-0009 MESHLESS SOLUTION OF INCOMPRESSIBLE FLOW OVER BACKWARD-FACING STEP Juraj MUŽÍK 1,* 1 Department of Geotechnics, Faculty of Civil Engineering, University

More information

The Method of Approximate Particular Solutions for Solving Elliptic Problems with Variable Coefficients

The Method of Approximate Particular Solutions for Solving Elliptic Problems with Variable Coefficients The Method of Approximate Particular Solutions for Solving Elliptic Problems with Variable Coefficients C.S. Chen, C.M. Fan, P.H. Wen Abstract A new version of the method of approximate particular solutions

More information

Using Modified Euler Method (MEM) for the Solution of some First Order Differential Equations with Initial Value Problems (IVPs).

Using Modified Euler Method (MEM) for the Solution of some First Order Differential Equations with Initial Value Problems (IVPs). Using Modified Euler Method (MEM) for the Solution of some First Order Differential Equations with Initial Value Problems (IVPs). D.I. Lanlege, Ph.D. * ; U.M. Garba, B.Sc.; and A. Aluebho, B.Sc. Department

More information

Lab - Introduction to Finite Element Methods and MATLAB s PDEtoolbox

Lab - Introduction to Finite Element Methods and MATLAB s PDEtoolbox Scientific Computing III 1 (15) Institutionen för informationsteknologi Beräkningsvetenskap Besöksadress: ITC hus 2, Polacksbacken Lägerhyddsvägen 2 Postadress: Box 337 751 05 Uppsala Telefon: 018 471

More information

Solution of Maximum Clique Problem. by Using Branch and Bound Method

Solution of Maximum Clique Problem. by Using Branch and Bound Method Applied Mathematical Sciences, Vol. 8, 2014, no. 2, 81-90 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.310601 Solution of Maximum Clique Problem by Using Branch and Bound Method Mochamad

More information

1.2 Numerical Solutions of Flow Problems

1.2 Numerical Solutions of Flow Problems 1.2 Numerical Solutions of Flow Problems DIFFERENTIAL EQUATIONS OF MOTION FOR A SIMPLIFIED FLOW PROBLEM Continuity equation for incompressible flow: 0 Momentum (Navier-Stokes) equations for a Newtonian

More information

Solutions of Stochastic Coalitional Games

Solutions of Stochastic Coalitional Games Applied Mathematical Sciences, Vol. 8, 2014, no. 169, 8443-8450 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.410881 Solutions of Stochastic Coalitional Games Xeniya Grigorieva St.Petersburg

More information

Contents. I The Basic Framework for Stationary Problems 1

Contents. I The Basic Framework for Stationary Problems 1 page v Preface xiii I The Basic Framework for Stationary Problems 1 1 Some model PDEs 3 1.1 Laplace s equation; elliptic BVPs... 3 1.1.1 Physical experiments modeled by Laplace s equation... 5 1.2 Other

More information

Study and implementation of computational methods for Differential Equations in heterogeneous systems. Asimina Vouronikoy - Eleni Zisiou

Study and implementation of computational methods for Differential Equations in heterogeneous systems. Asimina Vouronikoy - Eleni Zisiou Study and implementation of computational methods for Differential Equations in heterogeneous systems Asimina Vouronikoy - Eleni Zisiou Outline Introduction Review of related work Cyclic Reduction Algorithm

More information

Application of Finite Volume Method for Structural Analysis

Application of Finite Volume Method for Structural Analysis Application of Finite Volume Method for Structural Analysis Saeed-Reza Sabbagh-Yazdi and Milad Bayatlou Associate Professor, Civil Engineering Department of KNToosi University of Technology, PostGraduate

More information

A PRIMAL-DUAL EXTERIOR POINT ALGORITHM FOR LINEAR PROGRAMMING PROBLEMS

A PRIMAL-DUAL EXTERIOR POINT ALGORITHM FOR LINEAR PROGRAMMING PROBLEMS Yugoslav Journal of Operations Research Vol 19 (2009), Number 1, 123-132 DOI:10.2298/YUJOR0901123S A PRIMAL-DUAL EXTERIOR POINT ALGORITHM FOR LINEAR PROGRAMMING PROBLEMS Nikolaos SAMARAS Angelo SIFELARAS

More information

A Modified Spline Interpolation Method for Function Reconstruction from Its Zero-Crossings

A Modified Spline Interpolation Method for Function Reconstruction from Its Zero-Crossings Scientific Papers, University of Latvia, 2010. Vol. 756 Computer Science and Information Technologies 207 220 P. A Modified Spline Interpolation Method for Function Reconstruction from Its Zero-Crossings

More information

MODELING MIXED BOUNDARY PROBLEMS WITH THE COMPLEX VARIABLE BOUNDARY ELEMENT METHOD (CVBEM) USING MATLAB AND MATHEMATICA

MODELING MIXED BOUNDARY PROBLEMS WITH THE COMPLEX VARIABLE BOUNDARY ELEMENT METHOD (CVBEM) USING MATLAB AND MATHEMATICA A. N. Johnson et al., Int. J. Comp. Meth. and Exp. Meas., Vol. 3, No. 3 (2015) 269 278 MODELING MIXED BOUNDARY PROBLEMS WITH THE COMPLEX VARIABLE BOUNDARY ELEMENT METHOD (CVBEM) USING MATLAB AND MATHEMATICA

More information

Vertex Graceful Labeling of C j C k C l

Vertex Graceful Labeling of C j C k C l Applied Mathematical Sciences, Vol. 8, 01, no. 8, 07-05 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ams.01.5331 Vertex Graceful Labeling of C j C k C l P. Selvaraju 1, P. Balaganesan,5, J. Renuka

More information

Some New Generalized Nonlinear Integral Inequalities for Functions of Two Independent Variables

Some New Generalized Nonlinear Integral Inequalities for Functions of Two Independent Variables Int. Journal of Math. Analysis, Vol. 7, 213, no. 4, 1961-1976 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ijma.213.3485 Some New Generalized Nonlinear Integral Inequalities for Functions of

More information

Finite Element Convergence for Time-Dependent PDEs with a Point Source in COMSOL 4.2

Finite Element Convergence for Time-Dependent PDEs with a Point Source in COMSOL 4.2 Finite Element Convergence for Time-Dependent PDEs with a Point Source in COMSOL 4.2 David W. Trott and Matthias K. Gobbert Department of Mathematics and Statistics, University of Maryland, Baltimore County,

More information

Fatigue Crack Growth Simulation using S-version FEM

Fatigue Crack Growth Simulation using S-version FEM Copyright c 2008 ICCES ICCES, vol.8, no.2, pp.67-72 Fatigue Crack Growth Simulation using S-version FEM M. Kikuchi 1,Y.Wada 2, A. Utsunomiya 3 and Y. Li 4 Summary Fatigue crack growth under mixed mode

More information

A New Approach to Meusnier s Theorem in Game Theory

A New Approach to Meusnier s Theorem in Game Theory Applied Mathematical Sciences, Vol. 11, 2017, no. 64, 3163-3170 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.712352 A New Approach to Meusnier s Theorem in Game Theory Senay Baydas Yuzuncu

More information

Jacobi and Gauss Seidel Methods To Solve Elliptic Partial Differential Equations

Jacobi and Gauss Seidel Methods To Solve Elliptic Partial Differential Equations Jacobi and Gauss Seidel Methods To Solve Elliptic Partial Differential Equations Mohamed Mohamed Elgezzon Higher Institute for Engineering Technology-- Zliten mohamed_elgezzon@yahoo.com Abstract Solving

More information

Numerical Rectification of Curves

Numerical Rectification of Curves Applied Mathematical Sciences, Vol. 8, 2014, no. 17, 823-828 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.39500 Numerical Rectification of Curves B. P. Acharya, M. Acharya and S. B.

More information

PARALLEL METHODS FOR SOLVING PARTIAL DIFFERENTIAL EQUATIONS. Ioana Chiorean

PARALLEL METHODS FOR SOLVING PARTIAL DIFFERENTIAL EQUATIONS. Ioana Chiorean 5 Kragujevac J. Math. 25 (2003) 5 18. PARALLEL METHODS FOR SOLVING PARTIAL DIFFERENTIAL EQUATIONS Ioana Chiorean Babeş-Bolyai University, Department of Mathematics, Cluj-Napoca, Romania (Received May 28,

More information

ASSESSMENT OF COMPLEX VARIABLE BASIS FUNCTIONS IN THE APPROXIMATION OF IDEAL FLUID FLOW PROBLEMS

ASSESSMENT OF COMPLEX VARIABLE BASIS FUNCTIONS IN THE APPROXIMATION OF IDEAL FLUID FLOW PROBLEMS B. D. Wilkins, et al. Int. J. Comp. Meth. and Exp. Meas., Vol. 7, No. 1 (2019) 45 56 ASSESSMENT OF COMPLEX VARIABLE BASIS FUNCTIONS IN THE APPROXIMATION OF IDEAL FLUID FLOW PROBLEMS BRYCE D. WILKINS, T.V.

More information

Numerical Modelling in Fortran: day 6. Paul Tackley, 2017

Numerical Modelling in Fortran: day 6. Paul Tackley, 2017 Numerical Modelling in Fortran: day 6 Paul Tackley, 2017 Today s Goals 1. Learn about pointers, generic procedures and operators 2. Learn about iterative solvers for boundary value problems, including

More information

Department of Computing and Software

Department of Computing and Software Department of Computing and Software Faculty of Engineering McMaster University Assessment of two a posteriori error estimators for steady-state flow problems by A. H. ElSheikh, S.E. Chidiac and W. S.

More information

Ennumeration of the Number of Spanning Trees in the Lantern Maximal Planar Graph

Ennumeration of the Number of Spanning Trees in the Lantern Maximal Planar Graph Applied Mathematical Sciences, Vol. 8, 2014, no. 74, 3661-3666 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.44312 Ennumeration of the Number of Spanning Trees in the Lantern Maximal

More information

A numerical grid and grid less (Mesh less) techniques for the solution of 2D Laplace equation

A numerical grid and grid less (Mesh less) techniques for the solution of 2D Laplace equation Available online at www.pelagiaresearchlibrary.com Advances in Applied Science Research, 2014, 5(1):150-155 ISSN: 0976-8610 CODEN (USA): AASRFC A numerical grid and grid less (Mesh less) techniques for

More information

A Note on Vertex Arboricity of Toroidal Graphs without 7-Cycles 1

A Note on Vertex Arboricity of Toroidal Graphs without 7-Cycles 1 International Mathematical Forum, Vol. 11, 016, no. 14, 679-686 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/imf.016.667 A Note on Vertex Arboricity of Toroidal Graphs without 7-Cycles 1 Haihui

More information

INTERIOR POINT METHOD BASED CONTACT ALGORITHM FOR STRUCTURAL ANALYSIS OF ELECTRONIC DEVICE MODELS

INTERIOR POINT METHOD BASED CONTACT ALGORITHM FOR STRUCTURAL ANALYSIS OF ELECTRONIC DEVICE MODELS 11th World Congress on Computational Mechanics (WCCM XI) 5th European Conference on Computational Mechanics (ECCM V) 6th European Conference on Computational Fluid Dynamics (ECFD VI) E. Oñate, J. Oliver

More information

Efficient Second-Order Iterative Methods for IR Drop Analysis in Power Grid

Efficient Second-Order Iterative Methods for IR Drop Analysis in Power Grid Efficient Second-Order Iterative Methods for IR Drop Analysis in Power Grid Yu Zhong Martin D. F. Wong Dept. of Electrical and Computer Engineering Dept. of Electrical and Computer Engineering Univ. of

More information

The Paired Assignment Problem

The Paired Assignment Problem Open Journal of Discrete Mathematics, 04, 4, 44-54 Published Online April 04 in SciRes http://wwwscirporg/ournal/odm http://dxdoiorg/0436/odm044007 The Paired Assignment Problem Vardges Melkonian Department

More information

Journal of Engineering Research and Studies E-ISSN

Journal of Engineering Research and Studies E-ISSN Journal of Engineering Research and Studies E-ISS 0976-79 Research Article SPECTRAL SOLUTIO OF STEADY STATE CODUCTIO I ARBITRARY QUADRILATERAL DOMAIS Alavani Chitra R 1*, Joshi Pallavi A 1, S Pavitran

More information

A Computational Study on the Number of. Iterations to Solve the Transportation Problem

A Computational Study on the Number of. Iterations to Solve the Transportation Problem Applied Mathematical Sciences, Vol. 8, 2014, no. 92, 4579-4583 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.46435 A Computational Study on the Number of Iterations to Solve the Transportation

More information

Object Oriented Finite Element Modeling

Object Oriented Finite Element Modeling Object Oriented Finite Element Modeling Bořek Patzák Czech Technical University Faculty of Civil Engineering Department of Structural Mechanics Thákurova 7, 166 29 Prague, Czech Republic January 2, 2018

More information

What is Multigrid? They have been extended to solve a wide variety of other problems, linear and nonlinear.

What is Multigrid? They have been extended to solve a wide variety of other problems, linear and nonlinear. AMSC 600/CMSC 760 Fall 2007 Solution of Sparse Linear Systems Multigrid, Part 1 Dianne P. O Leary c 2006, 2007 What is Multigrid? Originally, multigrid algorithms were proposed as an iterative method to

More information

Rainbow Vertex-Connection Number of 3-Connected Graph

Rainbow Vertex-Connection Number of 3-Connected Graph Applied Mathematical Sciences, Vol. 11, 2017, no. 16, 71-77 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.612294 Rainbow Vertex-Connection Number of 3-Connected Graph Zhiping Wang, Xiaojing

More information

A new 8-node quadrilateral spline finite element

A new 8-node quadrilateral spline finite element Journal of Computational and Applied Mathematics 195 (2006) 54 65 www.elsevier.com/locate/cam A new 8-node quadrilateral spline finite element Chong-Jun Li, Ren-Hong Wang Institute of Mathematical Sciences,

More information

The 3D DSC in Fluid Simulation

The 3D DSC in Fluid Simulation The 3D DSC in Fluid Simulation Marek K. Misztal Informatics and Mathematical Modelling, Technical University of Denmark mkm@imm.dtu.dk DSC 2011 Workshop Kgs. Lyngby, 26th August 2011 Governing Equations

More information

Higher-Order Accurate Schemes for a Constant Coefficient Singularly Perturbed Reaction-Diffusion Problem of Boundary Layer Type

Higher-Order Accurate Schemes for a Constant Coefficient Singularly Perturbed Reaction-Diffusion Problem of Boundary Layer Type Applied Mathematical Sciences, Vol., 6, no. 4, 63-79 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.988/ams.6.639 Higher-Order Accurate Schemes for a Constant Coefficient Singularly Perturbed Reaction-Diffusion

More information

Sparse Grids. Lab 1. Discretization

Sparse Grids. Lab 1. Discretization Lab 1 Sparse Grids Lab Objective: Sparse Grids are an important tool when dealing with highdimensional problems. Computers operate in discrete space, not in continuous space. It is important to choose

More information

A mesh refinement technique for the boundary element method based on local error analysis

A mesh refinement technique for the boundary element method based on local error analysis A mesh refinement technique for the boundary element method based on local error analysis J. J. Rodriguez

More information

Lacunary Interpolation Using Quartic B-Spline

Lacunary Interpolation Using Quartic B-Spline General Letters in Mathematic, Vol. 2, No. 3, June 2017, pp. 129-137 e-issn 2519-9277, p-issn 2519-9269 Available online at http:\\ www.refaad.com Lacunary Interpolation Using Quartic B-Spline 1 Karwan

More information

Collocation and optimization initialization

Collocation and optimization initialization Boundary Elements and Other Mesh Reduction Methods XXXVII 55 Collocation and optimization initialization E. J. Kansa 1 & L. Ling 2 1 Convergent Solutions, USA 2 Hong Kong Baptist University, Hong Kong

More information

Triple Integrals in Rectangular Coordinates

Triple Integrals in Rectangular Coordinates Triple Integrals in Rectangular Coordinates P. Sam Johnson April 10, 2017 P. Sam Johnson (NIT Karnataka) Triple Integrals in Rectangular Coordinates April 10, 2017 1 / 28 Overview We use triple integrals

More information

Convergence of C 2 Deficient Quartic Spline Interpolation

Convergence of C 2 Deficient Quartic Spline Interpolation Advances in Computational Sciences and Technology ISSN 0973-6107 Volume 10, Number 4 (2017) pp. 519-527 Research India Publications http://www.ripublication.com Convergence of C 2 Deficient Quartic Spline

More information

New Basis Functions and Their Applications to PDEs

New Basis Functions and Their Applications to PDEs Copyright c 2007 ICCES ICCES, vol.3, no.4, pp.169-175, 2007 New Basis Functions and Their Applications to PDEs Haiyan Tian 1, Sergiy Reustkiy 2 and C.S. Chen 1 Summary We introduce a new type of basis

More information

THE GRAPH OF FRACTAL DIMENSIONS OF JULIA SETS Bünyamin Demir 1, Yunus Özdemir2, Mustafa Saltan 3. Anadolu University Eskişehir, TURKEY

THE GRAPH OF FRACTAL DIMENSIONS OF JULIA SETS Bünyamin Demir 1, Yunus Özdemir2, Mustafa Saltan 3. Anadolu University Eskişehir, TURKEY International Journal of Pure and Applied Mathematics Volume 70 No. 3 2011, 401-409 THE GRAPH OF FRACTAL DIMENSIONS OF JULIA SETS Bünyamin Demir 1, Yunus Özdemir2, Mustafa Saltan 3 1,2,3 Department of

More information

On a nested refinement of anisotropic tetrahedral grids under Hessian metrics

On a nested refinement of anisotropic tetrahedral grids under Hessian metrics On a nested refinement of anisotropic tetrahedral grids under Hessian metrics Shangyou Zhang Abstract Anisotropic grids, having drastically different grid sizes in different directions, are efficient and

More information

The MOMC Method: a New Methodology to Find. Initial Solution for Transportation Problems

The MOMC Method: a New Methodology to Find. Initial Solution for Transportation Problems Applied Mathematical Sciences, Vol. 9, 2015, no. 19, 901-914 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.4121013 The MOMC Method: a New Methodology to Find Initial Solution for Transportation

More information

FOURTH ORDER COMPACT FORMULATION OF STEADY NAVIER-STOKES EQUATIONS ON NON-UNIFORM GRIDS

FOURTH ORDER COMPACT FORMULATION OF STEADY NAVIER-STOKES EQUATIONS ON NON-UNIFORM GRIDS International Journal of Mechanical Engineering and Technology (IJMET Volume 9 Issue 10 October 2018 pp. 179 189 Article ID: IJMET_09_10_11 Available online at http://www.iaeme.com/ijmet/issues.asp?jtypeijmet&vtype9&itype10

More information

ACCELERATING CFD AND RESERVOIR SIMULATIONS WITH ALGEBRAIC MULTI GRID Chris Gottbrath, Nov 2016

ACCELERATING CFD AND RESERVOIR SIMULATIONS WITH ALGEBRAIC MULTI GRID Chris Gottbrath, Nov 2016 ACCELERATING CFD AND RESERVOIR SIMULATIONS WITH ALGEBRAIC MULTI GRID Chris Gottbrath, Nov 2016 Challenges What is Algebraic Multi-Grid (AMG)? AGENDA Why use AMG? When to use AMG? NVIDIA AmgX Results 2

More information

NIA CFD Seminar, October 4, 2011 Hyperbolic Seminar, NASA Langley, October 17, 2011

NIA CFD Seminar, October 4, 2011 Hyperbolic Seminar, NASA Langley, October 17, 2011 NIA CFD Seminar, October 4, 2011 Hyperbolic Seminar, NASA Langley, October 17, 2011 First-Order Hyperbolic System Method If you have a CFD book for hyperbolic problems, you have a CFD book for all problems.

More information

On the Parallel Implementation of Best Fit Decreasing Algorithm in Matlab

On the Parallel Implementation of Best Fit Decreasing Algorithm in Matlab Contemporary Engineering Sciences, Vol. 10, 2017, no. 19, 945-952 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ces.2017.79120 On the Parallel Implementation of Best Fit Decreasing Algorithm in

More information

COMPENDIOUS LEXICOGRAPHIC METHOD FOR MULTI-OBJECTIVE OPTIMIZATION. Ivan P. Stanimirović. 1. Introduction

COMPENDIOUS LEXICOGRAPHIC METHOD FOR MULTI-OBJECTIVE OPTIMIZATION. Ivan P. Stanimirović. 1. Introduction FACTA UNIVERSITATIS (NIŠ) Ser. Math. Inform. Vol. 27, No 1 (2012), 55 66 COMPENDIOUS LEXICOGRAPHIC METHOD FOR MULTI-OBJECTIVE OPTIMIZATION Ivan P. Stanimirović Abstract. A modification of the standard

More information

Research Article Data Visualization Using Rational Trigonometric Spline

Research Article Data Visualization Using Rational Trigonometric Spline Applied Mathematics Volume Article ID 97 pages http://dx.doi.org/.//97 Research Article Data Visualization Using Rational Trigonometric Spline Uzma Bashir and Jamaludin Md. Ali School of Mathematical Sciences

More information

A Comparative Study on Optimization Techniques for Solving Multi-objective Geometric Programming Problems

A Comparative Study on Optimization Techniques for Solving Multi-objective Geometric Programming Problems Applied Mathematical Sciences, Vol. 9, 205, no. 22, 077-085 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.2988/ams.205.42029 A Comparative Study on Optimization Techniques for Solving Multi-objective

More information

Comparison of different solvers for two-dimensional steady heat conduction equation ME 412 Project 2

Comparison of different solvers for two-dimensional steady heat conduction equation ME 412 Project 2 Comparison of different solvers for two-dimensional steady heat conduction equation ME 412 Project 2 Jingwei Zhu March 19, 2014 Instructor: Surya Pratap Vanka 1 Project Description The purpose of this

More information

Space-filling curves for 2-simplicial meshes created with bisections and reflections

Space-filling curves for 2-simplicial meshes created with bisections and reflections Space-filling curves for 2-simplicial meshes created with bisections and reflections Dr. Joseph M. Maubach Department of Mathematics Eindhoven University of Technology Eindhoven, The Netherlands j.m.l.maubach@tue.nl

More information

Numerical Methods to Solve 2-D and 3-D Elliptic Partial Differential Equations Using Matlab on the Cluster maya

Numerical Methods to Solve 2-D and 3-D Elliptic Partial Differential Equations Using Matlab on the Cluster maya Numerical Methods to Solve 2-D and 3-D Elliptic Partial Differential Equations Using Matlab on the Cluster maya David Stonko, Samuel Khuvis, and Matthias K. Gobbert (gobbert@umbc.edu) Department of Mathematics

More information

3D NURBS-ENHANCED FINITE ELEMENT METHOD

3D NURBS-ENHANCED FINITE ELEMENT METHOD 7th Workshop on Numerical Methods in Applied Science and Engineering (NMASE 8) Vall de Núria, 9 a 11 de enero de 28 c LaCàN, www.lacan-upc.es 3D NURBS-ENHANCED FINITE ELEMENT METHOD R. Sevilla, S. Fernández-Méndez

More information

Visualization of errors of finite element solutions P. Beckers, H.G. Zhong, Ph. Andry Aerospace Department, University of Liege, B-4000 Liege, Belgium

Visualization of errors of finite element solutions P. Beckers, H.G. Zhong, Ph. Andry Aerospace Department, University of Liege, B-4000 Liege, Belgium Visualization of errors of finite element solutions P. Beckers, H.G. Zhong, Ph. Andry Aerospace Department, University of Liege, B-4000 Liege, Belgium Abstract The aim of this paper is to show how to use

More information

A NEW HEURISTIC ALGORITHM FOR MULTIPLE TRAVELING SALESMAN PROBLEM

A NEW HEURISTIC ALGORITHM FOR MULTIPLE TRAVELING SALESMAN PROBLEM TWMS J. App. Eng. Math. V.7, N.1, 2017, pp. 101-109 A NEW HEURISTIC ALGORITHM FOR MULTIPLE TRAVELING SALESMAN PROBLEM F. NURIYEVA 1, G. KIZILATES 2, Abstract. The Multiple Traveling Salesman Problem (mtsp)

More information

Control Volume Finite Difference On Adaptive Meshes

Control Volume Finite Difference On Adaptive Meshes Control Volume Finite Difference On Adaptive Meshes Sanjay Kumar Khattri, Gunnar E. Fladmark, Helge K. Dahle Department of Mathematics, University Bergen, Norway. sanjay@mi.uib.no Summary. In this work

More information