Asymptotically exact a posteriori local discontinuous Galerkin error estimates for the one-dimensional second-order wave equation. (English) Zbl 1332.65134
The author is concerned with an implicit residual based a posteriori spatial error estimate for the semidiscrete local discontinuous Galerkin method related to the wave equation
\[
u_{tt}-u_{xx}=f(x,t),\quad (x,t)\in [-1,1]\times [0,T].
\]
The author obtains that the error estimates for the solution and its spatial derivative is of order \(O(h^{p+3/2})\), when \(p\) \(4\)-degree piecewise polynomials are used. The author also shows that the global effectivity indices for both the solution and its derivative in the \(L^2\) norm converge at the rate \(O(h^{1/2})\). Several numerical experiments are included in order to support the theoretical findings.
Reviewer: Marius Ghergu (Dublin)
MSC:
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 35L05 | Wave equation |
| 65M20 | Method of lines 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 |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
local discontinuous Galerkin method; second-order wave equation; superconvergence; residual-based a posteriori error estimates; semidiscrete; numerical experimentReferences:
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