Enhanced accuracy by post-processing for finite element methods for hyperbolic equations. (English) Zbl 1015.65049
The enhancement of accuracy, by means of a simple post-processing technique, for finite element approximations to transient hyperbolic equations is considered. The post-processing is a convolution with a kernel whose support has measure of order one in the case of arbitrary unstructured meshes; if the mesh is locally translation invariant, the support of the kernel is a cube whose edges are of size of the order of \(\Delta x\) only.
For example, when polynomials of degree \(k\) are used in the discontinuous Galerkin (DG) method, and the exact solution is globally smooth, the DG method is of order \(k+1/2\) in the \(L^2\)-norm, whereas the post-processed approximation is of order \(2k+1\); if the exact solution is in \(L^2 \) only, in which case no order of convergence is available for the DG method, the post-processed approximation converges with order \(k + 1/2\) in \(L^2 \left( {\Omega _0 }\right),\) where \(\Omega _0\) is a subdomain over which the exact solution is smooth. Numerical results displaying the sharpness of the estimates are presented.
For example, when polynomials of degree \(k\) are used in the discontinuous Galerkin (DG) method, and the exact solution is globally smooth, the DG method is of order \(k+1/2\) in the \(L^2\)-norm, whereas the post-processed approximation is of order \(2k+1\); if the exact solution is in \(L^2 \) only, in which case no order of convergence is available for the DG method, the post-processed approximation converges with order \(k + 1/2\) in \(L^2 \left( {\Omega _0 }\right),\) where \(\Omega _0\) is a subdomain over which the exact solution is smooth. Numerical results displaying the sharpness of the estimates are presented.
Reviewer: Leonid B.Chubarov (Novosibirsk)
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 35L65 | Hyperbolic conservation laws |
| 65M15 | Error bounds 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:
post-processing technique; enhancement of accuracy; finite element approximations; transient hyperbolic equations; discontinuous Galerkin method; convolution; convergence; numerical resultsReferences:
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