Numerical treatment for some abstract degenerate second-order evolutionary problem. (English) Zbl 1535.65179
Summary: This paper addresses the numerical analysis of a class of a degenerate second-order evolution equations. We employ a finite element method for spatial discretization and a family of implicit finite difference schemes for time discretization. Introducing a stabilization parameter, denoted by \(\theta\), we propose a well-posed fully-discrete scheme. Sufficient conditions for its well-posedness and for quasi-optimal error estimates are established. The abstract theory is illustrated through the application to the degenerate wave equation, and numerical results validate our theoretical findings.
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 |
| 76Q05 | Hydro- and aero-acoustics |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
| 35R09 | Integro-partial differential equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 74S20 | Finite difference methods applied to problems in solid mechanics |
| 35Q35 | PDEs in connection with fluid mechanics |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
Keywords:
second-order evolution equations; finite element method; error estimates; acoustic wave equationReferences:
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