An Interface-fitted Mesh Generator and Polytopal Element Methods for Elliptic Interface Problems

Transcription

1 An Interface-fitted Mesh Generator and Polytopal Element Methods for Elliptic Interface Problems Long Chen University of California, Irvine Joint work with: Huayi Wei (Xiangtan University), Min Wen(UCI) Polytopal Element Methods in Mathematics and Engineering Oct 27, 2015, Atlanta, GA

2 Outline Elliptic Interface Problems Interface-fitted Mesh Generation 2D Algorithm 3D Algorithm VEM for Elliptic Interface Problems Weak Formulation Virtual Element Method Weak Galerkin Methods Numerical Results

3 ELLIPTIC INTERFACE PROBLEMS

4 A Domain with an Interface Ω Ω n Ω + Γ Figure: A square domain Ω with an interface Γ in it.

5 Elliptic Interface Problems Consider (β u) = f, in Ω\Γ (1) with prescribed jump conditions across the interface Γ: [u] Γ = u + u = q 0, [βu n ] Γ = β + u + n β u n = q 1, and boundary condition u = g on Ω.

6 Existing Work Two type of numerical methods: Numerical methods based on Cartesian meshes. Numerical methods based on interface-fitted meshes.

7 Cartesian Mesh Approach Figure: A Catesian mesh and an interface.

8 Cartesian Mesh Approach Pro: Mesh generation is very simple Con: Need to modify the stencil (FDM) or basis function (FEM) of the vertex near the interface Γ. The linear system may be non-symmetric and cause problems in fast solvers. Convergence analysis is complicated.

9 Interface-fitted Mesh Figure: An interface-fitted mesh in 2D [Wei, Chen, Huang and Zheng, SISC. 2014].

10 Interface-fitted Mesh Approach Pro: Can use standard finite element methods to discretize the interface problem. Symmetric system can be easily solved by fast solver such as algebraic multigrid solvers. Relatively easy error analysis. Con: Generate interface-fitted meshes, extremely difficult in 3D!

11 Our Goal In the past decade the mesh generation have gotten great progress. Here we aim to 1. Develop effective and robust interface-fitted (polytopal) mesh generation algorithm. 2. Solve interface problems accurately using conforming FEM (e.g. Virtual Element method) on polytopal meshes. 3. Solve the resulting algebraic system quickly using (algebraic) multigrid solver.

12 Outline Elliptic Interface Problems Interface-fitted Mesh Generation 2D Algorithm 3D Algorithm VEM for Elliptic Interface Problems Weak Formulation Virtual Element Method Weak Galerkin Methods Numerical Results

13 INTERFACE-FITTED MESH GENERATION TWO DIMENSIONAL CASE

14 Delaunay Triangulation p 6 p 7 p 3 p 5 p 1 p 4 p 2 Figure: A Delaunay triangulation of a set of points is a triangulation satisfying the empty circle condition.

15 Delaunay Triangulation (a) Non Delaunay. (b) Delaunay. Figure: Empty circle condition: a circle circumscribing any Delaunay triangle does not contain any other input points in its interior.

16 2D Interface-fitted Mesh Generation Algorithm Algorithm 1 2D Interface-fitted Mesh Generation Algorithm INPUT: Grid size: h; Level set function, ϕ(x); Square domain, Ω; OUTOUT: Interface-fitted mesh T ; 1: Find the cut points, the Cartesian mesh points near or on the interface and some auxiliary points. 2: Construct a Delaunay triangulation on these points. 3: Post processing.

17 An Example in 2D: Step 1 Figure: Step 1: Find the cut points (red), the mesh points near or on the interface (black) and the auxiliary points (magenta).

18 An Example in 2D: Step 2 Figure: Step 2: Generate a Delaunay triangulation on these points. In MATLAB, it is simply DT = delaunay(x, y).

19 An Example in 2D: Step 3 Figure: Step 3: Keep the triangles in the interface elements.

20 An Example in 2D: Step 4 Figure: An interface-fitted mesh composed by triangles and squares.

21 Features of 2D Mesh Generator Simple, efficient and semi-unstructured. Recovery the interface. The maximum angle is uniformly bounded (135 ).

22 146 C. P aum Maximum Maximal interior Angle angle: u max Condition 90 triangle Kh;2 are smaller or equal 135. To prove this, note that 45 a3 90 and a1 60. This implies a2 180 a1 a Maximal interior angle: u max 135 Subdivision of Type d) boundary subcell: Let B be a Type d) boundary subcell. Then X \ B can be approximated by four triangles Kh;1, Kh;2, Kh;3 and Kh;4 such that a2 a1. The interior angles of the triangles Kh;1, Kh;2 and Kh;4 are less than or equal to 90. The interior angles of the triangle Kh;3 are less than or equal to 135. Semi-Unstructured This can Gridsbe proven as in the Type c) 147 subcell case. (a) θ max 90 (b) θ max 135. Maximal interior angle: u max 135 Subdivision of Type b) boundary subcell: Subdivision Let Maximal B be interior aof Type angle: b) c) boundary u max subcell. subcell: 135 Let B be a Type c) boundary subcell. Then X \ B can be approximated by twothen triangles X \ BKh;1 can andbekh;2. approximated Obviously, by thethree triangles Kh;1, Kh;2 and Kh;3. The interior angles of triangle Kh;1 are less than interior equal angles to 90of. the Thetriangles interior angles Kh;1 and of Kh;3 are less than or equal to 90. The triangle Kh;2 are smaller or equal 135. To interior proveangles this, note of thethat triangle 45 Kh;2 a3 are 90less than or equal to 135. To prove this, and a1 60. This implies a2 180 a1 note a3that 135 a3. 90 and b This implies a1 180 b Analogously, we obtain a Maximal interior angle: u max 135 (c) θ max 135. (d) θ max 135. Subdivision of Type d) boundary subcell: Subdivision Let B be aof Type interior d) boundary cells: Every subcell. interior cell can be subdivided into two triangles. triangles Kh;1, Kh;2, Kh;3 and Kh;4 such Then X \ B can be approximated by four that a2 a1. The interior angles of the triangles Kh;1, Kh;2 and Kh;4 are less than or equal to 90. The interior angles of the triangle Kh;3 are less than or equal to 135. This can be proven as in the Type c) subcell case. Subdivision of Type c) boundary subcell: Let B be a Type c) boundary subcell. Then X \ B can be approximated by three triangles Kh;1, Kh;2 and Kh;3. The Figure: Maximum angles of four types of interface elements. From Semi-Unstructured Grids by C. Pflaum. Computing 2001.

23 Interface Recovery Semi-Unstructured Grids 145 Figure 1. Di erent types of boundary subcells Figure 2. Boundary subcell of a double type

24 Interface Recovery Semi-Unstructured Grids 145 Figure 1. Di erent types of boundary subcells We DO NOT code the triangulation case by case. We simply call DT = delaunay(x, y). Figure 2. Boundary subcell of a double type

25 Interface Recovery Semi-Unstructured Grids 145 Figure 1. Di erent types of boundary subcells We DO NOT code the triangulation case by case. We simply call DT = delaunay(x, y). We can PROVE the interface will be preserved in the triangulation generated by delaunay and the maximal angle is minimized. Figure 2. Boundary subcell of a double type

26 Lower Convex Hull (a) Step 1: Lift points to the paraboloid (b) Step 2: Form the lowest convex hull in R n+1. (c) Step 3: Project the lowest convex hull to R n. Figure: Delaunay triangulation is the projection of the lower convex hull of points lifted to the paraboloid f = x 2.

27 Interface Recovery Semi-Unstructured Grids 145 Figure 1. Di erent types of boundary subcells The lower convex hull when lift to R n+1 will always connect the intersection points except...

28 A Degenerate Case A B C D Figure: Add an auxiliary point in a rectangle to preserve the interface

29 Efficiency Simple, Efficient and Semi-Unstructured. The greatest advantage: localization. Only call delaunay for O( N) points near the interface. The complexity will be thus O( N) in 2-D which can be ignored comparing with the O(N) complexity for assembling the matrix and solving the matrix equation. No mesh smoothing is needed. The maximal angle is bounded by 135. Good enough for finite element methods. The mesh is only unstructured near the interface and the majority of the mesh is structured which leads to nice properties (superconvergence, multigrid etc).

30 INTERFACE-FITTED MESH GENERATION THREE DIMENSIONAL CASE

31 Challenges in Tetrahedra Mesh Generation Figure: Tetrahedra elements [Shewchuk, 2002]. Most Tetrahedra mesh generation/optimization algorithms cannot guarantee sliver-free for a general 3D domain. Advertisement: Use ODT mesh smoothing which produces fewer slivers in 3D. L. Chen. Mesh smoothing schemes based on optimal Delaunay triangulations. 2004

32 3D Interface-fitted Mesh Generation Algorithm Algorithm 2 3D Interface-fitted Mesh Generation Algorithm INPUT: Grid size: h; Level set function, ϕ(x); Cube domain, Ω; OUTOUT: Interface-fitted mesh T ; 1: Find the cut points, the Cartesian mesh points near or on the interface and the auxiliary points. 2: Construct a Delaunay triangulation on these points. 3: Post processing: use polytopal meshes.

33 Polytopal Mesh v.s. Tetrahedron Mesh Figure: Bad tetrahedron elements will be eliminated and part of their faces will become the boundary of polytopes. As a surface mesh, the triangles are more acceptable.

34 Features of 3D Mesh Generator Simple, efficient and semi-unstructured. Recovery the interface. The maximum angle of surface mesh is uniformly bounded. Boundary faces of a polytope is either triangle or square.

35 Sphere Interface Figure: Sphere surface with 3, 128 triangles: maximum angle and minimum angle

36 Sphere Interface Figure: The mesh near the interface. Use second to generate a mesh with 70, 169 points).

37 Heart interface Figure: Heart surface with 3480 triangles: maximum angle and minimum angle 4.21.

38 Heart interface Figure: The mesh near the interface. Use second to generate a mesh with 70, 521 points).

39 Quartics interface Figure: Quartics surface with 44, 512 triangles: maximum angle and minimum angle 11.2.

40 Quartics interface Figure: The mesh near the interface. Use second to generate a mesh with 553, 161 points.

41 Torus interface Figure: Torus surface with 37, 600 triangles: maximum angle and minimum angle

42 Torus interface Figure: The mesh near the interface. Use second to generate a mesh with 1, 045, 061 points.

43 Outline Elliptic Interface Problems Interface-fitted Mesh Generation 2D Algorithm 3D Algorithm VEM for Elliptic Interface Problems Weak Formulation Virtual Element Method Weak Galerkin Methods Numerical Results

44 Elliptic Interface Problems Consider (β u) = f, in Ω\Γ (2) with prescribed jump conditions across the interface Γ: [u] Γ = u + u = q 0, [βu n ] Γ = β + u + n β u n = q 1, and boundary condition u = g on Ω.

45 Weak Formulation: Case 1 For the elliptic interface problem, if the jump conditions are homogeneous, i.e., q 0 = 0 and q 1 = 0 on Γ, then the problem is equivalent to that of finding u H 1 (Ω) with u = g on Ω such that β u vdx = fv dx, v H0 1 (Ω) (3) Ω Ω

46 Weak Formulation: Case 2 If q 0 = 0, q 1 0 on Γ, then the corresponding weak formulation is : (β u, v) Ω = (f, v) Ω q 1, v Γ, v H 1 0 (Ω). (4) The jump condition [βu n ] Γ = q 1 holds in H 1/2 (Γ) sense.

47 Weak Formulation: Case 3 If q 0 0 on Γ, we can find w : Ω R with w = q 0 on Ω and w H 1 (Ω ). The zero extension of w,denote by w satisfies w H 1 (Ω Ω + ) with w = w on Ω and w = 0 on Ω +. The choice of w is not unique. The interface problem is equivalent to: find u = p w with p H0 1 (Ω) such that: (β p, v) Ω = (f, v) Ω q 1, v Γ + (β w, v) Ω, v H 1 0 (Ω).

48 Virtual Element Method Virtual element method is a generalization of the standard conforming FEM on polygon or polyhedra meshes. L. Beirão Da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, and A. Russo. Basic principles of virtual element methods. Mathematical Models and Methods in Applied Sciences, 23(1): , B. Ahmad, A. Alsaedi, F. Brezzi, L. D. Marini, and A. Russo. Equivalent projectors for virtual element methods. Computers and Mathematics with Applications, 66(3): , Beirão da Veiga L., Brezzi F., Marini L.D. and Russo A. The Hitchhiker s Guide to the Virtual Element Method. Math. Models Meth. Appl. Sci. 24: (2014).

49 Spaces in Virtual Element Methods We introduce the following space on a simple polygon K V k (K) := {v H 1 (K) : v K B k ( K), v P k 2 (K)}, where P k (D) is the space of polynomials of degree k on D and conventionally P 1 (D) = 0, and the boundary space B k ( K) := {v C 0 ( K) : v e P k (e) for all edges e K}. VEM vs??? (recall that N K = nk + k(k 1)/2) VEM k =1 k =2 k =3

50 Weak Galerkin Methods Given a polygon mesh T h and an integer k 1, we introduce W h = {v = {v 0, v b }, v 0 K P k (K), v b e P k 1 (e) e K, K T h }. Define a modified weak gradient K w : W k (K) P k (K) as: given v = {v 0, v b } W k (K), define K w v P k (K) such that ( K w v, p) K = (v 0, p) K + v b, n p K for all p P k (K). L. Mu, J. Wang, and X. Ye. A weak Galerkin finite element method with polynomial reduction. Journal of Computational and Applied Mathematics, 285:45 58, 2015.

51 Equivalence The WG finite element is: find u h W 0 h such that a WG (u h, v) = (f, v 0 ) v = {v 0, v b } Wh 0. where a WG (u h, v h ) := ( w u h, w v h )+ χ b (u b u 0 K ) χ b (v b v 0 K ). K T h The non-conforming VEM is: find u h Ṽ 0 h such that ā VEM h (u h, v h ) = (f, Π 0 k v h) v h Ṽ 0 h, where ā VEM h (u, v) := ( Π k u, Π k v) + χ b (u Π 0 k u) χ b(v Π 0 k v). K T h

52 Outline Elliptic Interface Problems Interface-fitted Mesh Generation 2D Algorithm 3D Algorithm VEM for Elliptic Interface Problems Weak Formulation Virtual Element Method Weak Galerkin Methods Numerical Results

53 NUMERICAL RESULTS

54 Implementation and Errors Our numerical test was based on MATLAB package ifem. L. Chen. ifem: an integrated finite element method package in MATLAB We consider the following errors ( ) u I u h A = β 1/2 h (u I u h ) 2 + β 1/2 1/2 0,Ω h (u+ I u+ h ) 2 h 0,Ω + h u I u h = max( u I u h ) 0,,Ω, u + I u+ h h ) 0,,Ω + ) h ( ) 1/2 u I u h 0 = u I u h ) 2 + u + 0,Ω I u+ h ) 2 h 0,Ω + h where u h is the numerical approximation and u I is the nodal interpolation.

55 A Cube Domain with a Sphere Interface We chose the following setting φ(x, y, z) =x 2 + y 2 + z 2 r 2 u + =10(x + y + z) u =5e x2 +y 2 +z where r = The coefficient β is piecewise constant.

56 A Cube Domain with a Sphere Surface

57 Numerical Tests: β = 1 and β + = 1 #Dof h u I u h A u I u h u I u h u I u h C 1 h Error Error 10 2 Du I Du h C 1 h u I u h C 2 h log(1/h) 10 1 log(1/h)

58 Numerical Tests: β = 1 and β + = 10 #Dof h u I u h A u I u h u I u h u I u h C 1 h Error Error 10 2 Du I Du h C 1 h u I u h C 2 h log(1/h) 10 1 log(1/h)

59 Numerical Tests:β = 1 and β + = 100 #Dof h u I u h A u I u h u I u h u I u h C 1 h Error Error 10 2 Du I Du h C 1 h u I u h C 2 h log(1/h) 10 1 log(1/h)

60 Numerical Tests:β = 1 and β + = 1000 #Dof h u I u h A u I u h u I u h u I u h Error 10 1 Error 10 2 C 1 h Du I Du h C 1 h u I u h C 2 h log(1/h) 10 1 log(1/h)

61 Numerical Tests: β = 1 and β + = #Dof h u I u h A u I u h u I u h u I u h Du I Du h 10 2 C 1 h Error 10 1 C 1 h u I u h Error C 2 h log(1/h) 10 1 log(1/h)

62 Algebraic Multigrid Solvers #Dof β + = 1 β + = 10 β + = 100 β + = 1000 β + = Table: Iteration steps of MGCG using W-cycle and AMG preconditioner with fixed β = 1 and increased β +.

63 Summary Simple, Efficient and Semi-Unstructured 2-D and 3D Interface-fitted Mesh Generator.

64 THANK YOU FOR YOUR ATTENTION!

65 THANK YOU FOR YOUR ATTENTION! Supported by NSF DMS