On some numerical convergence studies of mixed finite element methods for flow in porous media

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1 On some numerical convergence studies of mixed finite element methods for flow in porous media Gergina Pencheva Abstract We consider an expanded mixed finite element method for solving second-order elliptic partial differential equations. We study the effects of nonmatching grids, discontinuous coefficients, and high variation in the coefficients on the accuracy of the numerical solution. The error in the case of nonmatching grids and smooth solutions occurs mainly along the interfaces and high accuracy is preserved in the interior. Discontinuous coefficients may lead to singular solutions and the polution from the singularity affects the accuracy in the whole domain. Our last set of examples shows that the dependence of the convergence rates and constants in front of the error terms on high variation in the coefficients is very weak. 1 Introduction In this work we consider mixed finite element method for subdomain discretizations. Mixed methods owe their popularity to their local (element-wise) mass conservation property and the simultaneous and accurate approximation of two variables of physical interest, e.g., pressure and velocity in fluid flow. In many applications the complexity of the geometry or the behavior of the solution prompts the use of multiblock domain structure where the simulation domain is decomposed into a series of nonoverlapping subdomains (blocks). Each block is independently covered by a local grid. A non-overlpping domain decomposition algorithm was developed for matching grids by Glowinski and Wheeler [5, 3] ans was later extended to non-matching grids. Mortar finite elements are used to impose physically meaningfull matching conditions on the interfaces while mixed finite elements are applied locally on the subdomains (see [6, 1] for details). In this work we consider a second-order elliptic equation which in porous medium applications models single phase Darsy flow. We solve for the pressure p and the velocity field u satisfying u = K p in Ω, (1) u + αp = f in Ω, (2) p = g D on Γ D, (3) u ν = g N on Γ N, (4) where α 0 represents the rock compressibility; Ω R d, d = 2 or 3 is a multiblock domain; K is symmetric, uniformly positive definite tensor with smooth or perhaps piecewise smooth components representing the permeability divided by the viscosity; ν is outward unit normal vector on Ω; and Ω is decomposed into Γ D and Γ N. The problem was solved using the parallel domain decomposition code Parcel [4] with some modifications made by the author. The code implements an expanded mixed finite element method developed by Arbogast, Wheeler and Yotov [2] where mixed method with tensor coefficient is writen as a cell-centered finite difference method by incorporating certain quadrature rules. In the case of nonmatching grids we study the convergence of interior velocity (far from subdomain interfaces). The results show that the interior velocity error is superconvergent of O(h 2 ), which means that majority 1

2 of the error occurs near the interfaces. Therefore we need to apply some local postprocessing to obtain better convergence rate for the velocity error. Second group of tests was run in the case of discontinuous tensor for both mathing and nonmatching grids. As the results show, because of the strong singularity at the cross-point (1/2, 1/2), there is no superconvergence even in the interior. The maximum rate of convergence for the interior velocity error is of O(h). Therefore to control the error we need some local refinement near this cross-point. Analyzing all test results in group 1 and group 2 we may conclude that interior velocity error depends on the smoothness of the solution in the whole domain Ω, but in a more weak sense, and that interior velocity error is better than the velocity error calculated over the whole domain. The last group of tests studies the influence of the the low order term α in (2) on the constant C in the error estimate p p h Ch 2. We compared the results when α = 0 (no low order term) and α = 1. The results show that this method works very well for both cases even if there are big variations of K and that the constant increases very slowly when the ratio goes up. The rest of the paper is organized as follows. Interior error estimates in the case of nonmatching grids are presented in Section 2. Error estimates in the case of discontinuous tensor are presented in Section 3. In section 4 the influence of the low order term on the constant in the error estimates is studied. 2 Interior error estimates in the case of nonmatching grids To study the interior velocity error we used six tests with known analytic solutions. All examples are on the unit cube. The domain is divided into four subdomains with interfaces along the x = 1/2 and y = 1/2 lines. The boundary conditions are Dirichlet on the left and right face and Neumann on the rest of the boundary. In test#58 we have p(x, y, z) = x 3 y 2 + sin(xy) and K = cos(xy) In test#59 p(x, y, z) = cos( πx 2 ) cos(πy ) and K = I. 2 In test#64 we have a problem with discontinuous coefficient { I, 0 x < 1/2 K = 10 I, 1/2 < x 1. The solution x 2 y 3 + cos(xy), 0 x < 1/2 p(x, y, z) = ( ) 2x y 3 + cos( 2x+9 20 y), 1/2 < x 1 is chosen to be continuous and to have continuous normal flux at x = 1/2. In the next three tests K is a full tensor. In test#104 In test#107 p(x, y, z) = x + y + z 1.5 and K =. x 2 + y z sin(xy) 0 sin(xy) x 2 y p(x, y, z) = x 2 (x 1) 2 y 2 (y 1) 2 z 2 (z 1) 2 and K =

3 Finally in test#110 and { xy, 0 x 1/2 p(x, y, z) = xy + (x 1/2)(y + 1/2), 1/2 x , 0 x < 1/2 K = I, 1/2 < x 1 For the 2d-problems (## 58, 59, 64, 110) the initial nonmatching grids are given in Figure 1 and the initial mortar grids on all interfaces are given in Table 1. For 3d-problems (#104, #107) we consider similar (but 3d ) grids. We use α = 0.1. Figure 1: Initial non-matching grids for Cases 1 3 mortar elements Table 1: Initial number of elements in mortar grids for Cases Case 1 The code was first modified to calculate the error over the interiors of all subdomains that have a one-element border around it. Tests were only run using mortar 4 (piecewise constant) because it was obvious that even there exist some improvement of the the rate of convergence of the interior error, it is not essential. The results for this case are in Table 2. The rates were established by running all tests for 5 levels of grid refinement (we halve the element diameters for each refinement) and computing a least squares fit to the error. 2.2 Case 2 The second modification of the code produced a scaled interior error. The calculation u u h / u over the interior subdomains was used in an attempt to eliminate any possible influence in the size of the interior subdomains would have on the error calculations. Again the improvement wasn t essential. The results for this case are in Table 3. 3

4 velocity L 2 error vel. L 2 err. Int. velocity L error vel. L err. Int. test C u α u C u α u C u α u C u α u Table 2: Velocity errors for Case 1 mortar 4 mor velocity L 2 error vel. L 2 err. Int. velocity L error vel. L err. Int. tar test C u α u C u α u C u α u C u α u Table 3: Velocity errors for Case 2 4

5 2.3 Case 3 Thirdly, the code was modified to calculate the errors over fixed interior domains for each level of refinement. In this case it seems that the interior velocity error is superconvergent of O(h 2 ), which means that majority of the error occurs near the interfaces. Therefore we need to apply some local postprocessing to obtain better convergence rate for the velocity error. The results for this case are in Table 4 and Table 5. Plots of the computed solution and the numerical error for the case of mortar 4 are shown in Figure 2 through Figure 7. mor flux error pressure L 2 error λ error tar test C f α f C p α p C λ α λ Table 4: Velocity errors for Case 3 Part I 5

6 mor velocity L 2 error vel. L 2 err. Int. velocity L error vel. L err. Int. tar test C u α u C u α u C u α u C u α u Table 5: Velocity errors for Case 3 Part II 6

7 A. Computed pressure and velocity B. Pressure and velocity error Figure 2: Solution and error (magnified) for test#58 mortar4 A. Computed pressure and velocity B. Pressure and velocity error Figure 3: Solution and error (magnified) for test#59 mortar4 7

8 A. Computed pressure and velocity B. Pressure and velocity error Figure 4: Solution and error (magnified) for test#64 mortar4 A. Computed pressure and velocity B. Pressure and velocity error Figure 5: Solution and error (magnified) for test#104 mortar4 8

9 A. Computed pressure and velocity B. Pressure and velocity error Figure 6: Solution and error (magnified) for test#107 mortar4 A. Computed pressure and velocity B. Pressure and velocity error Figure 7: Solution and error (magnified) for test#110 mortar4 9

10 3 Error estimates in the case of discontinuous tensor Because it is hard to find problems with discontinuous tensor and known true solution for which the right-hand side f is a smooth function, we needed to make a bigger modification in the code. Thus, first the code was run for the finest grid and the solution was saved in files. Then the stored solution from this initial run was used to calculate the errors for all coarser grids. Again all examples are on the unit cube; the domain was divided into four equal subdomains. The initial grid in the case of matching grids was chosen to be The initial nonmatching and mortar grids are shown in Table Non-matching grids Table 6: Initial grids for Case 4 mortar elements Mortar grids In this case different test problems were tested. In problems 70 through 75 the permeability tensors were diagonal with piecewise constant diagonal elements. The prototype for the permeability tensor is K = a(x, y) a(x, y) a(x, y) where 10 n, if x < 1/2, y < 1/2 a(x, y) = 10 n, if x > 1/2, y > 1/2 1, otherwise For test problem#70,n = 1 and then n increments by 1 with each test problem through test#73. problem# , x < 1/2, y < 1/2 a(x, y) = 10, x > 1/2, y > 1/2 1, otherwise For test problem#75 10, x < 1/2, y < 1/2 a(x, y) = 10 2, x > 1/2, y > 1/2 1, otherwise Test problem#170 is with full tensor a(x, y).1a(x, y) 0 K =.1a(x, y) a(x, y) a(x, y) For test where a(x, y) is as in test problem#70. The boundary conditions are Dirichlet on the left and right face and Neumann (no flow) on the rest of the boundary. For test problems#70,#71,#72, #73,#170 p x=0 = 1 while for test problems#74 and #75 p x=0 = 10 10

11 For all tests p x=1 = 0 The results for this case are in Table 7 and Table 8. Plots of the computed solution and the numerical error for test problems#71 and #170 are shown in Figure 8 and Figure 9. As the results show, because of the strong singularity at the cross-point (1/2, 1/2), there is no superconvergence even in the interior. The maximum rate of convergence for the interior velocity error is of O(h). Therefore to control the error we need some local refinement near this cross-point. Conclusion:Analyzing all test results in Section 2 and Section 3 we may conclude that interior velocity error depends on the smoothness of the solution in the whole domain Ω, but in more weak sense, and that interior velocity error is better than the velocity error calculated over the whole domain. mortar 4 mortar 3 mortar 2 mortar 1 matching grids flux error pressure L 2 error λ error test C f α f C p α p C λ α λ Table 7: Errors for Case 4 Part I 11

12 mortar 4 mortar 3 mortar 2 mortar 1 matching grids velocity L 2 error vel. L 2 err. Int. velocity L error vel. L err. Int. test C u α u C u α u C u α u C u α u Table 8: Errors for Case 4 Part II 12

13 A. Computed pressure and velocity B. Pressure and velocity error Figure 8: Solution and error (magnified) for test#71 matching grids A. Computed pressure and velocity B. Pressure and velocity error Figure 9: Solution and error (magnified) for test#170 matching grids 13

14 4 Influence of the low order term on the constant in the error estimates Theory indicates that the constant C in the error estimate p p h Ch 2 depends on K max /K min. We study the dependence of the constant on the low order term α in (2). We tested three groups of problems. For all of them the permeability tensor K was chosen to be diagonal matrix with diagonal elements a(x) = e β(x 1 2 )2 where β is a real, nonnegative parameter. The values of β and corresponding values (approximately) of the ratio K max /K min are given in Table 9. β ratio Table 9: Values of Kmax K min Test problems#81 and #82 used true analytic solutions. For test#81 p = 1 x and for test#82 p = x 3 y 4 + x 2 + sin(xy) cos(y) For test#83 we used again files to save the solution for the finest grid. For this test f 0, p x=0 = 1, p x=1 = 0 and u ν = 0 on Γ N. For all test problems we used matching grids and the boundary conditions were Dirichlet on the left and right face and Neumann on the rest of the boundary. We compared the results when α = 0 (no low order term) and α = 1. The results are in Table 10 through Table 15.. Plots of the computed solution and the numerical error for α = 0 and α = 1 are shown in Figure 10 through Figure 15. They show that this method works very well for both cases even if there are big variations of K and that the constant increases very slowly when the ratio goes up. flux error pressure L 2 error λ error α C f α f C p α p C λ α λ E E E E E E ratio Table 10: Errors for Test#81 Part I 14

15 velocity L 2 error vel. L 2 err. Int. velocity L error vel. L err. Int. α C u α u C u α u C u α u C u α u E E E E E E E E ratio Table 11: Errors for Test#81 Part II flux error pressure L 2 error λ error ratio α C f α f C p α p C λ α λ Table 12: Errors for Test#82 Part I velocity L 2 error vel. L 2 err. Int. velocity L error vel. L err. Int. ratio α C u α u C u α u C u α u C u α u Table 13: Errors for Test#82 Part II 15

16 flux error pressure L 2 error λ error ratio α C f α f C p α p C λ α λ E E E E E E Table 14: Errors for Test#83 Part I velocity L 2 error vel. L 2 err. Int. velocity L error vel. L err. Int. ratio α C u α u C u α u C u α u C u α u E E E E E E E E Table 15: Errors for Test#83 Part II A. Computed pressure and velocity B. Pressure and velocity error Figure 10: Solution and error (magnified) for test#81 α = 0 β = 46 matching grids 16

17 A. Computed pressure and velocity B. Pressure and velocity error Figure 11: Solution and error (magnified) for test#81 α = 1 β = 46 matching grids A. Computed pressure and velocity B. Pressure and velocity error Figure 12: Solution and error (magnified) for test#82 α = 0 β = 46 matching grids 17

18 A. Computed pressure and velocity B. Pressure and velocity error Figure 13: Solution and error (magnified) for test#82 α = 1 β = 46 matching grids A. Computed pressure and velocity B. Pressure and velocity error Figure 14: Solution and error (magnified) for test#83 α = 0 β = 46 matching grids 18

19 References A. Computed pressure and velocity B. Pressure and velocity error Figure 15: Solution and error (magnified) for test#83 α = 1 β = 46 matching grids [1] T. ARBOGAST, L. C. COWSAR, M. F. WHEELER, AND I. YOTOV, Mixed finite element methods on nonmatching multiblock grids, SIAM J. Numer. Anal., 37 (2000), pp [2] T. ARBOGAST, M. F. WHEELER, AND I. YOTOV, Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences, SIAM J. Numer. Anal., 34 (1997), pp [3] L C. COWSAR AND M. F. WHEELER, Parallel domain decomposition method for mixed finite elements for elliptic partial differential equations, in Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, [4] L C. COWSAR, C A. SAN SOUCIE, AND I YOTOV, Parcel v1.04 User Guide, May [5] R. GLOWINSKI AND M. F. WHEELER, Domain decomposition and mixed finite element methods for elliptic problems, in First International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, 1988, pp [6] I. YOTOV, Mixed finite element methods for flow in porous media, PhD Thesis, Rice University, Houston, Texas. TR96-09, Dept. Comp. Appl. Math., Rice University and TICAM report 96-23, University of Texas at Austin. 19