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Finite element approximation of reaction-diffusion problems using an exponentially graded mesh. (English) Zbl 1442.65355

Summary: We present the analysis of an \(h\) version Finite Element Method for the approximation of the solution to singularly perturbed reaction-diffusion problems posed in smooth domains \(\Omega\subset\mathbb R^2\). The method uses piecewise polynomials of degree \(p\) in each variable, defined on an exponentially graded mesh, optimally constructed for the approximation of exponential layers. We establish robust, optimal convergence rates in a variety of norms and illustrate our theoretical findings through numerical computations.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
35B25 Singular perturbations in context of PDEs
35J25 Boundary value problems for second-order elliptic equations

Software:

DistMesh
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Full Text: DOI

References:

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