Mixed virtual element method for integro-differential equations of parabolic type. (English) Zbl 07895356
Summary: This article presents and analyzes a mixed virtual element approach for discretizing parabolic integro-differential equations in a bounded subset of \(\mathbb{R}^2\), in addition to the backward Euler approach for temporal discretization. With the help of the intermediate projection along with Fortin and \(L^2\) projections, we effectively tackle the treatment of integral terms in both the fully discrete and semi-discrete analysis. This inclusion leads to the derivation of optimal a priori error estimates with an order of \(O(h^{k+1})\) for the two unknowns. Furthermore, we present a systematic analysis that outlines the step-by-step process for achieving super convergence of the discrete solution, with an order of \(O(h^{k+2})\). Several computational experiments are discussed to validate the proposed scheme’s computational efficiency and support the theoretical conclusions.
MSC:
| 65R20 | Numerical methods for integral equations |
| 45K05 | Integro-partial differential equations |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
mixed virtual element method (VEM); parabolic integro-differential equations; error estimates; super convergenceReferences:
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