Supercloseness analysis of a stabilizer-free weak Galerkin finite element method for viscoelastic wave equations with variable coefficients. (English) Zbl 1514.65131
Summary: In this article, we are concerned about a stabilizer-free weak Galerkin (SFWG) finite element method for approximating a second-order linear viscoelastic wave equation with variable coefficients. For SFWG solutions, both semidiscrete and fully discrete convergence analysis is considered. The second-order Newmark scheme is employed to develop the fully discrete scheme. We obtain supercloseness of order two, which is two orders higher than the optimal convergence rate in \(L^{\infty}(L^2)\) and \(L^{\infty}(H^1)\) norms. In other words, we attain \(\mathcal{O}(h^{k+3}+\tau^2)\) in \(L^{\infty}(L^2)\) norm and \(\mathcal{O}(h^{k+2}+\tau^2)\) in \(L^{\infty}(H^1)\) norm. Several numerical experiments in a two-dimensional setting are carried out to validate our theoretical convergence findings. These experiments confirm the robustness and accuracy of the proposed method.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 76A10 | Viscoelastic fluids |
| 86A15 | Seismology (including tsunami modeling), earthquakes |
| 76Q05 | Hydro- and aero-acoustics |
| 74D05 | Linear constitutive equations for materials with memory |
Keywords:
viscoelastic wave equations; stabilizer-free weak Galerkin method; semidiscrete and fully discrete schemes; superclosenessReferences:
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