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Singh, Gautam; Natesan, Srinivasan; Sendur, Ali

Superconvergence error analysis of discontinuous Galerkin method with interior penalties for 2D elliptic convection-diffusion-reaction problems. (English) Zbl 1524.65858

Int. J. Comput. Math. 100, No. 5, 948-967 (2023).
Summary: This article focuses on constructing and analyzing a non-symmetric interior penalty Galerkin method (NIPG) on Shishkin mesh for solving singularly perturbed 2D elliptic boundary-value problems (BVPs). We use piecewise Lagrange interpolation at Gaussian points to improve the order of convergence of the interpolation error. We then study the superconvergence properties of the NIPG method and prove \(\mathcal{O}(N^{-1}\ln N)^{k+1}\) order of convergence in the discrete energy-norm. Various numerical experiments are provided to validate the theoretical results.

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
65N15 Error bounds for boundary value problems involving PDEs
35B25 Singular perturbations in context of PDEs
65D05 Numerical interpolation
35J25 Boundary value problems for second-order elliptic equations

Keywords:

singularly perturbed 2D elliptic BVPs; convection-diffusion problem; Shishkin mesh; discontinuous Galerkin method with interior penalty; superconvergence
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References:

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