×

A priori estimates for the solution of convection-diffusion problems and interpolation on Shishkin meshes. (English) Zbl 0892.35014

Summary: The solution of singularly perturbed convection-diffusion problems can be split into a regular and a singular part containing the boundary layer terms. In dimensions \(n=1\) and \(n=2\), sharp estimates of the derivatives of both parts up to order 2 are given. The results are applied to estimate the interpolation error for the solution on Shishkin meshes for piecewise bilinear finite elements on rectangles and piecewise linear elements on triangles. Using the anisotropic interpolation theory it is proved that the interpolation problem on Shishkin meshes is quasi-optimal in \(L_\infty\) and in the energy norm.

MSC:

35B25 Singular perturbations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
65D05 Numerical interpolation
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Apel, T. and M. Dobrowolski: Anisotropic interpolation with applications to the finite element method. Computing 47 (1992), 277 - 293. · Zbl 0746.65077 · doi:10.1007/BF02320197
[2] Kellogg, R. B. and A. Tsan: Analysis of some difference approximations for a singularly perturbed two-point boundary value problem. Math. Comp. 32 (1978), 1025 - 1039. · Zbl 0418.65040 · doi:10.2307/2006331
[3] Kellogg, R. B.: Boundary layers and corner singularities for a self-adjoint problem: In: Boundary Value Problems and Integral Equations in Non-Smooth Domains (eds.: M. Costabel et al.). New York: Dekker 1995, 121 - 149. · Zbl 0824.35006
[4] Miller, J. J. H., O’Riordan, E. and C. I. Shishkin: Fitted numerical methods for singular perturbation problems. Singapore: World Scientific 1996. · Zbl 0915.65097
[5] Roos, H.-G., Stynes, M. and L. Tobiska: Numerical Methods for Singularly Perturbed Differential Equations. Heidelberg: Springer- Verlag 1996. · Zbl 0844.65075
[6] Roos, H.-G., Adam, D. and A. Felgenhauer: A novel uniformly convergent finite element method in two dimensions. J. Math. Anal. AppI. 201 (1996), 711 - 755. · Zbl 0859.65118 · doi:10.1006/jmaa.1996.0283
[7] Shishkin, C. I.: Discrete Approximation of Singularly Perturbed Elliptic and Parabolic Equations (in Russian). Ekaterinburg: Russ. Acad. Sci. 1992.
[8] Schwab, C., Sun, M. and C. Xenophontos: The hp finite element method for problems in mechanics with boundary layers. Report. ETH Zärich: Report 96-20 (1996).
[9] Stynes, M. and E. O’Riordan: A uniformly convergent Galerkin method on a Shishkin mesh for a convection-diffusion problem. J. Math. Anal, and AppI. (to appear). · Zbl 0917.65088 · doi:10.1006/jmaa.1997.5581
[10] 2eni9 ek, A. and M. Vanmaele: The interpolation theorem for narrow quadrilateral isopara- metric elements. Numer. Math. 72 (1995), 123 - 141. · Zbl 0839.65005 · doi:10.1007/s002110050163
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.