zbMATH — the first resource for mathematics

The combination technique for a two-dimensional convection-diffusion problem with exponential layers. (English) Zbl 1212.65443
Summary: Convection-diffusion problems posed on the unit square and with solutions displaying exponential layers are solved using a sparse grid Galerkin finite element method (FEM) with Shishkin meshes. Writing $$N$$ for the maximum number of mesh intervals in each coordinate direction, our “combination” method simply adds or subtracts solutions that have been computed by the Galerkin FEM on $$N \times \sqrt N$$, $$\sqrt N \times N$$ and $$\sqrt N \times \sqrt N$$ meshes. It is shown that the combination FEM yields (up to a factor $$\ln N$$) the same order of accuracy in the associated energy norm as the Galerkin FEM on an $$N\times N$$ mesh, but it requires only $$\mathcal O(N^{3/2})$$ degrees of freedom compared with the $$\mathcal O(N^2)$$ used by the Galerkin FEM. An analogous result is also proved for the streamline diffusion finite element method.

MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs 65F10 Iterative numerical methods for linear systems
Full Text:
References:
 [1] R. E. Bank: Hierarchical bases and the finite element method. Acta Numerica 5 (1996), 1–43. · Zbl 0865.65078 [2] H.-J. Bungartz, M. Griebel: Sparse grids. Acta. Numerica 13 (2004), 147–269. · Zbl 1118.65388 [3] H.-J. Bungartz, M. Griebel, U. Rüde: Extrapolation, combination, and sparse grid techniques for elliptic boundary value problems. Comput. Methods Appl. Mech. Eng. 116 (1994), 243–252. · Zbl 0824.65104 [4] P. G. Ciarlet: The Finite Element Method for Elliptic Problems. SIAM, Philadelphia, 2002. [5] F.-J. Delvos: d-Variate Boolean interpolation. J. Approximation Theory 34 (1982), 99–114. · Zbl 0504.41004 [6] M. Dobrowolski, H.-G. Roos: A priori estimates for the solution of convection-diffusion problems and interpolation on Shishkin meshes. Z. Anal. Anwend. 16 (1997), 1001–1012. · Zbl 0892.35014 [7] J. Garcke, M. Griebel: On the computation of the eigenproblems of hydrogen and helium in strong magnetic and electric fields with the sparse grid combination technique. J. Comput. Phys. 165 (2000), 694–716. · Zbl 0979.65101 [8] M. Griebel, M. Schneider, C. Zenger: A combination technique for the solution of sparse grid problem. Iterative Methods in Linear Algebra. Proceedings of the IMASS, International symposium, Brussels, Belgium, April 2–4, 1991 (P. de Groen, R. Beauwens, eds.). North-Holland, Amsterdam, 1992, pp. 263–281. [9] Q. Lin, N. Yan, A. Zhou: A sparse finite element method with high accuracy I. Numer. Math. 88 (2001), 731–742. · Zbl 0989.65134 [10] T. Linß: Uniform superconvergence of a Galerkin finite element method on Shishkin-type meshes. Numer. Methods Partial Differ. Equations 16 (2000), 426–440. · Zbl 0958.65110 [11] T. Linß: Layer-adapted meshes for convection-diffusion problems. Comput. Methods Appl. Mech. Eng. 192 (2003), 1061–1105. · Zbl 1022.76036 [12] T. Linß, M. Stynes: Asymptotic analysis and Shishkin-type decomposition for an elliptic convection-diffusion problem. J. Math. Anal. Appl. 261 (2001), 604–632. · Zbl 1200.35046 [13] F. Liu, N. Madden, M. Stynes, A. Zhou: A two-scale sparse grid method for a singularly perturbed reaction-diffusion problem in two dimensions. IMA J. Numer. Anal. To appear. · Zbl 1188.65153 [14] F. Liu, A. Zhou: Two-scale finite element discretizations for partial differential equations. J. Comput. Math. 24 (2006), 373–392. · Zbl 1100.65101 [15] F. Liu, A. Zhou: Localizations and parallelizations for two-scale finite element discretizations. Commun. Pure Appl. Anal. 6 (2007), 757–773. · Zbl 1141.65079 [16] F. Liu, A. Zhou: Two-scale Boolean Galerkin discretizations for Fredholm integral equations of the second kind. SIAM J. Numer. Anal. 45 (2007), 296–312. · Zbl 1144.65087 [17] J. J. Miller, E. O’Riordan, G. I. Shishkin: Fitted Numerical Methods for Singular Perturbation Problems. World Scientific, Singapore, 1996. [18] J. Noordmans, P. W. Hemker: Application of an additive sparse-grid technique to a model singular pertubation problem. Computing 65 (2000), 357–378. · Zbl 0986.65118 [19] E. O’Riordan, G. I. Shishkin: A technique to prove parameter-uniform convergence for a singularly perturbed convection-diffusion equation. J. Comput. Appl. Math. 206 (2007), 136–145. · Zbl 1117.65145 [20] C. Pflaum, A. Zhou: Error analysis of the combination technique. Numer. Math. 84 (1999), 327–350. · Zbl 0942.65122 [21] H.-G. Roos, M. Stynes, L. Tobiska: Numerical Methods for Singularly Perturbed Differential Equations. Springer, Berlin, 1996. [22] M. Stynes, E. O’Riordan: A uniformly convergent Galerkin method on a Shishkin mesh for a convection-diffusion problem. J. Math. Anal. Appl. 214 (1997), 36–54. · Zbl 0917.65088 [23] M. Stynes, L. Tobiska: The SDFEM for a convection-diffusion problem with a boundary layer: Optimal error analysis and enhancement of accuracy. SIAM J. Numer. Anal. 41 (2003), 1620–1642. · Zbl 1055.65121 [24] J. Xu: Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal. 33 (1996), 1759–1777. · Zbl 0860.65119 [25] H. Yserentant: On the multi-level splitting of finite element spaces. Numer. Math. 49 (1986), 379–412. · Zbl 0608.65065 [26] C. Zenger: Sparse grids. In: Parallel Algorithms for Partial Differential Equations (Proc. 6th GAMM-Seminar, Kiel, 1990). Notes Numer. Fluid Mech. 31. 1991, pp. 241–251. [27] Z. Zhang: Finite element superconvergence on Shishkin mesh for 2-D convection-diffusion problems. Math. Comput. 72 (2003), 1147–1177. · Zbl 1019.65091
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.