FOR ALL GRID SIZES. Thor Gjesdal. Christian Michelsen Research A/S. N-5036 Fantoft, Norway SUMMARY

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1 A CELL-CENTERED MULTIGRID ALGORITHM FOR ALL GRID SIZES Thor Gjesdal Christian Michelsen Research A/S N-5036 Fantoft, Norway SUMMARY Multigrid methods are optimal; that is, their rate of convergence is independent of the number of grid points, because they use a nested sequence of coarse grids to represent dierent scales of the solution. This nesting does, however, usually lead to certain restrictions of the permissible size of the discretised problem. In cases where the modeler is free to specify the whole problem, such constraints are of little importance because they can be taken into consideration from the outset. We consider the situation in which there are other competing constraints on the resolution. These restrictions may stem from the physical problem (e.g., if the discretised operator contains experimental data measured on a xed grid) or from the need to avoid limitations set by the hardware. In this paper we discuss a modication to the cell-centered multigrid algorithm, so that it can be used for problems with any resolution. We discuss in particular a coarsening strategy and choice of intergrid transfer operators that can handle grids with both an even or odd number of cells. The method is described and applied to linear equations obtained by discretisation of two- and three-dimensional second-order elliptic PDEs. INTRODUCTION Multigrid methods have during the last decades developed into an important tool in many areas of scientic computation. Because they use a nested sequence of grids to represent dierent scales of the solution, the multigrid algorithms are optimal in the sense that their computational complexity is linearly proportional to the total number of unknowns in the discretised problem. This nesting does however lead to certain restrictions on the permissible size of the discrete problem, similar to those encountered in other ecient `divide-and-conquer' algorithms such as the fast Fourier transform or cyclic reduction. In a standard multigrid algorithm, coarsening is usually This work was supported by the Research Council of Norway through Grant number /410 and program number STP{30074.

2 performed by doubling the mesh-spacing. The number of cells in the grid will then be of the form n l = C l 2 k, where n l is the number of cells in direction l and C l is some suitable (small) integer. Early applications of multigrid methods for general grid sizes consisted of padding the ne grid with empty cells. Such padding can lead to potential large overheads in storage requirements and computational complexity. Dendy [1] and Adams [2] have both described modications of vertex-centered multigrid algorithms that are extended to handle general grid-sizes. In cases with an odd number of cells, Dendy employs a dummy point on the coarse grid, while Adams has devised a special coarsening strategy using a uniform grid spacing at all levels. Often the modeler is free to specify the whole problem, and then such constraints are of no importance because they can be taken into consideration from the outset. We consider here the situation in which there are other competing constraints on the resolution. These restrictions may stem from the physical problem (e.g., if the discretised operator contains experimental data measured on a xed grid) or from the need to avoid limitations set by the hardware. We believe that these restrictions must be overcome if the multigrid methods are ever to become a standard inventory in the modeler's toolbox. In this paper we discuss a modication to the cell-centered multigrid algorithm, so that it can be used for problems with any resolution. The cell-centered algorithm is attractive because cell-centered discretisations are in widespread use, and cellcentered multigrid also has the ability to handle problems with discontinuous or rapidly varying diusion coecients using standard grid transfer operators [3, 4, 5]. In the next section we will describe the method with special emphasis on the grid coarsening and construction of the intergrid transfer functions. We will apply the method to linear equations obtained by discretisation of two- and three-dimensional second-order elliptic PDEs and show that the convergence rates are indeed independent of the grid size (even grids with an odd number of cells). MULTIGRID ALGORITHM Two level algorithm To describe the method, we will consider a two-level algorithm for the discretised problem Au = b; where A is the discretised dierential operator, which we assume is linear. The two-level algorithm consists of a smoothing step and a correction step where the update to the solution is calculated on a coarse grid. The two components of the multigrid algorithm are complementary; that is, smoothing is used to reduce high frequency error components, while the coarse grid correction is good at eliminating low frequencies in the error. We will denote coarse grid quantities by an overbar, and we can then write the algorithm in symbolic form as M = S 2 (I? P A?1 RA)S 1 ;

3 Level nx ny Figure 1: Multigrid hierarchy for ve-level system to illustrate grid coarsening strategy. where M is the two-level error reduction operator; R; P are the restriction and prolongation operators, and S is the smoothing operator with 1 ; 2 the number of preand post-smoothing sweeps, respectively. We obtain the multigrid algorithm by recursive application of the two-level algorithm to solve the coarse level defect equation Ae = Rr = R(b? Au). Grid coarsening For a given ne grid, we choose the coarse grid size as n l = bn l =2c + mod(n l ; 2) (1) where bc is the oor function. Standard coarsening, or coarsening in all coordinate directions, is performed as far as possible. For rectangular grids, semi-coarsening is then continued until the coarsest grid has a small number of cells in each direction. In this way, a coarse grid hierarchy is dened for any ne grid, and multigrid iterations can be performed. To illustrate the coarsening strategy, gure 1 shows an example of two ve-level systems. The rst example shows the standard case with a suitable number of cells and full coarsening in four levels. In the second example we have `bad' numbers (an odd number of cells in both directions and a moderately rectangular grid). Then we apply full coarsening for two levels and continue semi-coarsening for two levels to obtain a small system on the coarsest grid. Transfer operators In this section will we describe the restriction and prolongation operators. For simplicity we will concentrate on the one-dimensional operators. We will then describe briey how we obtain the higher dimensional operators. The grid transfer operators must satisfy the well-known accuracy requirement m R + m P > 2M; where m R and m P are the order of the restriction and the prolongation, respectively, and 2M is the order of the dierential operator. The order of the grid transfer operators and this rule can be determined either by considering how the interpolation acts

4 on the Fourier components (Brandt [6], Hemker [7]) or the order of the polynomials used in the interpolation rule (Hackbusch [8]). If we consider second order elliptic operators, we will use a restriction based on linear interpolation, which gives m R = 2, and for the prolongation we will use piecewise constant interpolation (m P = 1). This seems to be more robust than the opposite alternative (m R = 1; m P = 2) [9]. Prolongation, or coarse-to-ne interpolation, is performed by cell-based piecewise constant interpolation; that is, the coarse grid function values are transferred directly to the ne-grid points that belong to each coarse grid cell. The ne-to-coarse restriction is dened by the average u i = (Ru) i = X j R(i; j)u i+j ; = dn=ne : (2) The one-dimensional restriction operator is given by the adjoint of linear interpolation. In the standard case, where n is an even number, this restriction is simply given by the stencil h i R = : In general, when n is either odd or even, we can envisage two methods to construct the restriction operator. First, we can adopt an approach akin to that of Adams [2] and assume that both the coarse and the ne computational grids are given as a uniform distribution of cells on the unit interval, with spacing h = 1=n. The restriction weights can then be calculated by R(i; j) = max 0; 1? (2i + j? 1)h? (i? 1)h)=h 2 2 : (3) This will give a three- or four-point stencil in all cases. These restriction weights should be scaled so that they add up to the ratio n=n [10]. If the ne grid has an even number of cells, this formula will reproduce the standard stencil. This approach will unfortunately not produce a stable coarse-grid operator when we use the Galerkin approximation, and as a consequence the convergence rate of the method will deteriorate. We have therefore instead developed a restriction operator based on true cell-based coarsening. In this case, one cell at the boundary will be identical to the boundary cell at the ner level, as illustrated in gure 2. A similar aproach was suggested by Hutchinson and Raithby [11] in connection with the use of a low-order restriction operator. For a restriction based on the adjoint of linear interpolation we must modify the stencil in the cells close to the boundary. We get 1 3 R(n? 1; :) = [ 1? w 0 ]; 4 4 R(n; :) = [ w ]; where w = a=b. If k is the number of immediately preceding ner levels that has an odd number of cells, a and b are given by a 1 = 1 a k = 2a k?1 ; b 1 = 3 b k = 2b k?1? 1:

5 1/3 1 Fine Grid Coarse grid Figure 2: Fine and coarse cells at boundary when the ne grid has an odd number of cells. The numbers indicate the restriction weights for the end point. When semicoarsening is used, a direction exists in which n = n. In this case = 1, and both the restriction and the prolongation are given by the identity operator I = [ ]. In the multidimensional case, the stencils for restriction and prolongation are determined by tensor products of the one-dimensional stencils. If we let i and j denote multi-indices, we will for example have for the restriction stencil in 3D R 3D (i; j) = R x (i 1 ; j 1 )R y (i 2 ; j 2 )R z (i 3 ; j 3 ): (4) In other words, in two- and three dimensions restriction will be given by the adjoint of bi- and tri-linear interpolation, respectively. Coarse grid approximation There coarse grid matrices are determined by the use of the Galerkin approximation A = RAP. The Galerkin coarse grid approximation is preferable to straightforward discretisation, because the coarse grid operator can be automatically calculated from the ne grid stencils. This can give the multigrid solver the appearance of a black-box solver where the user only has to supply the coecients of the discretised equations. Because the restriction operator is based on bi- and tri-linear interpolation in higher dimensions, the coarse grid stencil will be full (9 points in 2D, 27 points in 3D.) The stencil elements can readily be calculated by the algorithm given by Wesseling [10]. Smoothing A point that should be noted is that the coarsening strategy we described in the previous section may change the (an-)isotropy of the coarse-grid operator compared

6 to the operator on the ne grid. This might have to be taken into consideration when we select the smoother. In two dimensions the alternating line Gauss-Seidel method is a robust smoother that is not too expensive. In practice its performance is often quite comparable to Red-Black point relaxation even for isotropic operators. In 3D, the only really robust smoother is alternating plane relaxations, which unfortunately is rather expensive even if a multigrid method is used, to solve the two-dimensional planes. It is therefore dicult to recommend this smoother without reservation. If we have enough knowledge of the problem at hand to decide that a line relaxation method would suce, a potential gain can be harvested, but for a truly black-box solution plane relaxation is probably the safest bet. Implementation aspects In this section we will discuss briey some practical aspects of the algorithm. The use of one-dimensional interpolation rules makes the implementation fairly modular, and by using features such as derived data types and dynamic memory allocation that are now available in Fortran we have written a combined two- and three-dimensional solver where the PDE can be discretised on any compact stencil (the most common are 5, 7, or 9 points in 2D and 7, 12, 19, or 27 points in 3D). With the use of recursion, it is also possible that the solver calls itself for plane smoothing in 3D problems. Depending on how restriction and prolongation are treated, a modest overhead related to the transfer operations will be realized. In our implementation, this overhead is on the order of nx ny nz oating point operations (roughly equivalent to one residual calculation on the ne grid) per iteration and nx+ny+nz memory locations. The overhead related to work can however be eliminated if all stencil elements are precomputed and stored. This option will of course lead to a larger storage penalty. COMPUTATIONAL EXAMPLES In this section we will demonstrate the convergence of the method for some selected test examples. Laplace/Poisson equation First we consider the Laplace equation on rectangular regions with Dirichlet or Neumann boundary conditions. r 2 u = 0; x 2 u = 1; x or = 0; x

7 Table 1: Two-Dimensional Laplace Equation with Dirichlet Boundary Conditions Grid size Levels Reduction factor Iterations D and 3D calculations with a uniform ne grid The rst set of calculations is performed on the Laplace equation with Dirichlet boundary conditions in a case in which the grid spacing is the same in each direction, so that the ne-grid operator is isotropic. The iterations start o from random initial values in the unknowns and are performed until the residual norm is reduced by a factor of 10?12. The average residual reduction rate,, is dened as = krn k 2 kr 0 k 2! 1=n : Table 1 shows results of two-dimensional calculations using alternating line Gauss- Seidel as the smoother for a series of dierent grid sizes. The results are given for a V(0,1) (sawtooth) cycle. We see from the table that the method works equally well for problem sizes that include both `good' and `bad' multigrid numbers. Results of three-dimensional calculations are given in table 2. The results indicate that the alternating line smoother is suciently robust to handle cases in which either odd-numbered grids or semi-coarsening lead to anisotropy in the coarse grid problems Table 2: Three-Dimensional Laplace Equation with Dirichlet Boundary Conditions Grid size Line GS Plane GS Reduction Iterations time Reduction Iterations time

8 Table 3: Two-Dimensional Laplace Equation with Homogeneous Neumann Boundary Conditions on a Stretched Grid Stretching factor: Grid size n n n Diverge as long as the problem on the ne grid is isotropic. The dramatic slow-down seen in the case where we use alternating plane relaxation may be caused by the start-up overhead of the multigrid solver. A way to alleviate this might be to rework the plane-smoother to precompute the coecients in all the planes. This will however lead to a considerable storage overhead. Another alternative is to investigate whether the three-dimensional coecients that are already computed can be used also in the plane solver. Examples with a non-uniform grid In this section, we will study the eect of anisotropy by introducing a nonuniform grid. Botta and Wubs [12] have shown that solution of partial dierential equations on a stretched grid can pose a challenging test case for iterative methods. One of their test cases consists of the two-dimensional Laplace equation on the unit square with homogeneous Neumann boundary conditions. The initial eld is given by f = x 2 (1? y) 2, and the convergence criterion used is that the absolute error should be below 10?6. The grid is generated by geometric stretching so that s = h i+1 =h i is constant. We use a V(0,1) cycle and the alternating line smoother; the results are shown in table 3. We note that the method fails to converge for large values of the stretching factor. The experiments indicate that a critical stretching factor exists depending on the grid size, and that iterations will diverge if the stretching is greater than this limit. In practice will we however only encounter moderate stretching, because an appreciable loss of accuracy occurs even for stretching factors larger than, say, s 5=4. We also performed the same experiment in a 3D case with a 32 3 grid, and we noticed that for moderate stretching rates, s 2, we obtained essentially no degradation in the convergence using the alternating line smoother. For s = 5, we did, however, notice a signicant slow-down as expected. Stone's problem This problem was introduced by Stone [13] as a test case for the Strongly Implicit Procedure (SIP), which is a relaxation method based on an incomplete LU decomposition. The problem consists of a heterogeneous diusion problem on the unit square

9 Source = 0.5 Kx = Ky = 0 Sink = Kx = Ky = 1 Sink = Kx = 100 Ky = 1 Kx = 1 Ky = 100 Source = 1.0 Source = 0.6 given by Figure 3: Geometry for Stone's model problem. r diag(k x ; K y )ru ~n ru? = f; (x; y) 2 [0; 1] 2 ; = 0: The geometry of the problem, specifying the conductivities and the sources, is depicted in gure 3. This problem was solved on a grid with cells, using 4 levels in the multigrid iterations. The initial eld was identically zero. The convergence factors for this problem are given in table 4. SIMPLE pressure correction equation The pressure correction equation in the SIMPLE algorithm for solution of the incompressible Navier-Stokes equations can be interpreted as an elliptic operator in Table 4: Stone's Model Problem ( = 10?8 ) Cycle Reduction factor Iterations V(0,1) V(1,1)

10 Table 5: Pressure Correction Equation in the Two Backward Facing Step Examples; Results for the First 20 Outer Iterations BFS BFS-POR Iteration n n Average the form D(Gp 0 ) = Du; where D and G are discretised divergence and gradient operators, respectively, p 0 is the pressure correction and u is the intermediate velocity eld. The diusivity consists of the inverse of a diagonalisation of the momentum operator and geometric terms. The equation is usually employed with homogeneous Neumann boundary conditions. The multigrid solver has been used to solve the pressure correction equation for the test case of a backward-facing step at Re = 800. The step size was half the channel height, and the channel had a length of 30 step heights. The grid had cells, which gives a grid aspect ratio of 5 : 1. In the rst test, the standard set-up from the benchmark results of Gartling [14] was used. In order to assess the eect of the use of porosities on convergence rates, we performed a second test in which a thin vertical plate with zero porosity and a height equal to the step height was placed in the channel downstream of the two main recirculation bubbles. Table 5 shows the residual reduction rates for the multigrid solver in the two examples. The results show that even though the convergence of the solver deteriorates in the case for which we have zero porosity (and as a consequence, zero diusivity ), its performance is still good. CONCLUSIONS We have described a generalisation of the cell-centered multigrid algorithm to cover problems with general resolution. Smoothing and the multigrid scheduling are not aected by the extensions, but changes have been made in the grid coarsening strategy and, consequently, in the design of the intergrid transfer operators. We found that cell-based coarsening was better than a mapping of the coarse grid to a uniform mesh. Numerical experiments show that the solver gives multigrid convergence for all

11 grid sizes in a number of test cases. The alternating line Gauss-Seidel relaxation is a good smoother for the two-dimensional solver. Its performance was also satisfactory in some 3D problems. In some cases involving extreme grid stretching the method seems to fail. This failure is, however, of small practical importance. REFERENCES [1] Dendy, J. E., Black box multigrid, J. Comput. Phys., 48:366{386, [2] Adams, J. C., MUDPACK 2: multigrid software for approximating elliptic partial dierential equations on uniform grids with any resolution, Appl. Math. Comput., 53:235{249, [3] Wesseling, P., Cell-centered multigrid for interface problems, J. Comput. Phys., 79:85{91, [4] Khalil, M., Analysis of Linear Multigrid Methods for Elliptic Dierential Equations with Discontinuous and Anisotropic Coecients, Ph.D. thesis, Delft University of Technology, [5] Khalil, M. and Wesseling, P., Vertex-centered and cell-centered multigrid for interface problems, J. Comput. Phys., 98:1{20, [6] Brandt, A., Guide to multigrid development, in Hackbusch, W. and Trottenberg, U., editors, Multigrid Methods, volume 960 of Lecture Notes in Mathematics, pp. 220{312, Springer-Verlag, Berlin, [7] Hemker, P. W., On the order of prolongations and restrictions in multigrid procedures, J. Comput. Appl. Math., 32:423{429, [8] Hackbusch, W., Multigrid Methods and Applications, volume 4 of Computational Mathematics, Springer{Verlag, Berlin, [9] Ersland, B. G. and Teigland, R., Comparison of two cell{centered multigrid schemes for problems with discontinuous coecients, Numer. Meth. for PDE, 9:265{283, [10] Wesseling, P., An Introduction to Multigrid Methods, Pure and Applied Mathematics, John Wiley & Sons, Chichester, [11] Hutchinson, B. R. and Raithby, G. D., A Multigrid Method Based on the Additive Correction Strategy, Numer. Heat Transf., 9:511{537, [12] Botta, E. F. F. and Wubs, F. W., The convergence behaviour of iterative methods on severely stretched grids, Int. J. Numer. Meth. Engng., 36:3333{3350, 1993.

12 [13] Stone, H. L., Iterative Solution of Implicit Approximations of Multidimensional Partial Dierential Equations, SIAM J. Numer. Anal., 5:530{558, [14] Gartling, D. K., A Test Problem for Outow Boundary Conditions Flow over a Backward-Facing Step, Int. J. Numer. Methods Fluids, 11:953{967, 1990.

SELECTIVE ALGEBRAIC MULTIGRID IN FOAM-EXTEND

Student Submission for the 5 th OpenFOAM User Conference 2017, Wiesbaden - Germany: SELECTIVE ALGEBRAIC MULTIGRID IN FOAM-EXTEND TESSA UROIĆ Faculty of Mechanical Engineering and Naval Architecture, Ivana