Abstract
We consider the numerical solution of a two-dimensional singularly perturbed reaction-diffusion problem posed on the unit square by a multiscale sparse grid finite element method. A Shishkin mesh which resolves the boundary and corner layers, and yields a parameter robust solution, is used. Our analysis shows that the method achieves essentially the same level of accuracy, in the energy norm, as the standard Galerkin finite element method with bilinear elements. However, only \(\mathcal {O}(N\log N)\) degrees of freedom are required, compared to \(\mathcal {O}(N^{2})\) for the corresponding Galerkin finite element method. This may be regarded as a generalisation of Liu et al. (IMA J. Numer. Anal. 29(4), 986–1007 2009) which used a two-scale method requiring \(\mathcal {O}(N^{3/2})\) degrees of freedom. Numerical results are provided that demonstrate the sharpness of the estimates and the efficiency of the method.
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References
Apel, T.: Anisotropic finite elements: local estimates and applications. Advances in Numerical Mathematics. B. G. Teubner, Stuttgart (1999)
Bungartz, H.-J., Griebel, M.: Sparse grids. Acta Numer. 13, 147–269 (2004)
Chegini, N., Stevenson, R: The adaptive tensor product wavelet scheme: sparse matrices and the application to ingularly perturbed problems. IMA J. Numer. Anal. 32(1), 75–104 (2012)
Chen, Y., Davis, T.A., Hager, W.W., Rajamanickam, S.: Algorithm 887: CHOLMOD, supernodal sparse Cholesky factorization and update/downdate. ACM Trans. Math. Softw. (2008). doi: 10.1145/1391989.1391995
Ciarlet, P.G.: The finite element method for elliptic problems. North-Holland Publishing Co., Amsterdam. Studies in Mathematics and its Applications, vol. 4 (1978)
Clavero, C., Gracia, J. L., O’Riordan, E: A parameter robust numerical method for a two dimensional reaction-diffusion problem. Math. Comp. 74(252), 1743–1758 (2005)
Franz, S., Liu, F., Roos, H.-G., Stynes, M., Zhou, A: The combination technique for a two-dimensional convection-diffusion problem with exponential layers. Appl. Math. 54(3), 203–223 (2009)
Gracia, J. L., Clavero, C.: A compact finite difference scheme for 2D reaction-diffusion singularly perturbed problems. J. Comput. Appl. Math. 192(1), 152–167 (2006)
Griebel, M., Zenger, C., Zimmer, S.: Multilevel Gauss-Seidel-algorithms for full and sparse grid problems. Computing 50(2), 127–148 (1993)
Griebel, M., Schneider, M., Zenger, C.: A combination technique for the solution of sparse grid problems. In: Iterative methods in linear algebra (Brussels, 1991), pp 263–281. Amsterdam, North-Holland (1992)
Han, H., Kellogg, R. B.: Differentiability properties of solutions of the equation −𝜖 2Δu+ru=f(x,y) in a square. SIAM J. Math. Anal. 21(2), 394–408 (1990)
Hegland, M., Garcke, J., Challis, V.: The combination technique and some generalisations. Linear Algebra Appl. 420(2-3), 249–275 (2007)
Kellogg, R.B., Linß, T., Stynes, M.: A finite difference method on layer-adapted meshes for an elliptic reaction-diffusion system in two dimensions. Math. Comp. 77(264), 2085–2096 (2008)
Lin, R., Stynes, M.: A balanced finite element method for singularly perturbed reaction-diffusion problems. SIAM J. Numer. Anal. 50(5), 2729–2743 (2012)
Linß, T.: Layer-adapted meshes for reaction-convection-diffusion problems. Lecture notes in Mathematics 1985, vol. xi, p 320. Springer, Berlin (2010)
Liu, F., Madden, N., Stynes, M., Zhou, A.: A two-scale sparse grid method for a singularly perturbed reaction-diffusion problem in two dimensions. IMA J. Numer. Anal. 29(4), 986–1007 (2009)
MacLachlan, S., Madden, N.: Robust Solution of Singularly Perturbed Problems Using Multigrid Methods. SIAM J. Sci. Comput. 35(5), A2225–A2254 (2013)
Miller, J. J. H., O’Riordan, E., Shishkin, G. I.: Fitted numerical methods for singular perturbation problems. World Scientific Publishing Co. Inc., River Edge, NJ (1996)
Noordmans, J., Hemker, P. W.: Application of an adaptive sparse-grid technique to a model singular perturbation problem. Computing 65(4), 357–378 (2000)
Pflaum, C., Zhou, A.: Error analysis of the combination technique. Numer. Math. 84(2), 327–350 (1999)
Roos, H.-G., Schopf, M.: Convergence and stability in balanced norms of finite element methods on Shishkin meshes for reaction-diffusion problems. ZAMM, Z. Angew. Math. Mech (Forthcoming)
Roos, H.-G., Stynes, M., Tobiska, L.: Robust numerical methods for singularly perturbed differential equations, vol. 24 of Springer Series in Computational Mathematics, Second Edition. Springer-Verlag, Berlin (2008)
Russell, S., Madden, N.: An introduction to the analysis and implementation of sparse grid finite element methods. http://www.maths.nuigalway.ie/~niall/SparseGrids/ (2014)
Yserentant, H.: On the multilevel splitting of finite element spaces. Numer. Math. 49(4), 379–412 (1986)
Christoph Zenger: Sparse grids. In: Parallel algorithms for partial differential equations (Kiel, 1990), volume 31 of Notes Numer. Fluid Mech., pp 241–251. Vieweg, Braunschweig (1991)
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Communicated by: M. Stynes
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Madden, N., Russell, S. A multiscale sparse grid finite element method for a two-dimensional singularly perturbed reaction-diffusion problem. Adv Comput Math 41, 987–1014 (2015). https://doi.org/10.1007/s10444-014-9395-7
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DOI: https://doi.org/10.1007/s10444-014-9395-7