Error estimates for the standard Galerkin-finite element method for the shallow water equations

Antonopoulos, D.; Dougalis, V.

doi:10.1090/mcom3040

Error estimates for the standard Galerkin-finite element method for the shallow water equations
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by D. C. Antonopoulos and V. A. Dougalis PDF

Math. Comp. 85 (2016), 1143-1182 Request permission

Abstract:

We consider a simple initial-boundary-value problem for the shallow water equations in one space dimension and also the analogous problem for a symmetric variant of the system. Assuming smoothness of solutions, we discretize these problems in space using standard Galerkin-finite element methods and prove $L^{2}$-error estimates for the semidiscrete problems for quasiuniform and uniform meshes. In particular we show that in the case of spatial discretizations with piecewise linear continuous functions on a uniform mesh, suitable compatibility conditions at the boundary and superaccuracy properties of the $L^{2}$ projection on the finite element subspaces lead to an optimal-order $O(h^{2})$ $L^{2}$-error estimate. We also examine the temporal discretization of the semidiscrete problems by a third-order explicit Runge-Kutta method due to Shu and Osher and prove $L^{2}$-error estimates of optimal order in the temporal variable under a Courant-number stability condition. In a final section of remarks we prove optimal-order $L^{2}$-error estimates for smooth spline spatial discretizations of the periodic initial-value problem for the systems. We also prove that small-amplitude, appropriately transformed solutions of the symmetric system are close to the corresponding solutions of the usual system while they are both smooth, thus providing a justification of the symmetric system.

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