Analysis of an LDG-FEM for a two-dimensional singularly perturbed convection-reaction-diffusion problem with interior and boundary layers. (English) Zbl 07710408
Summary: This article is concerned with a two-dimensional singularly perturbed convection-reaction-diffusion interface problem. In the considered problem, the convection and reaction coefficients and the source term have discontinuities along interface lines, which are parallel to the \(x\)- and \(y\)-axes. The coefficient of the highest-order term is a small positive parameter denoted by \(\varepsilon\). Due to discontinuities in the coefficients and the source term, interior and boundary layers appear in the solution when \(\varepsilon\) approaches zero. A Local Discontinuous Galerkin method is constructed on an appropriate Shishkin mesh. The test functions in the Local Discontinuous Galerkin method are piecewise polynomials that lie in the space \(\mathcal{Q}_r\) of piecewise polynomials of degree at most \(r\) in each variable, where \(r\) is a positive integer. We established that the error in the computed solution converges at the rate of \(\mathcal{O}((N^{-1}\ln N)^{r+\frac{1}{2}})\) in an energy-norm. Numerical results are given to support the theoretical results.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 35K57 | Reaction-diffusion equations |
| 35B25 | Singular perturbations in context of PDEs |
Keywords:
singular perturbation; convection-reaction-diffusion; interior layers; corner layers; boundary layers; Shishkin mesh; finite element method; LDG methodReferences:
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