A priori and a posteriori error estimates of a $\mathcal {C}^0$-in-time method for the wave equation in second order formulation

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2026年8月5日

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Z. Dong,
L. Mascotto,
Z. Wang

摘要

We establish fully-discrete a priori and semi-discrete in time a posteriori error estimates for a discontinuous-continuous Galerkin discretization of the wave equation in second order formulation; the resulting method is a Petrov–Galerkin scheme based on piecewise polynomial test functions and continuous piecewise polynomial trial functions in time, respectively. Crucial tools in the a priori analysis for the fully-discrete formulation are the design of suitable projection and interpolation operators extending those used in the parabolic setting, and stability estimates based on a nonstandard choice of the test function; a priori estimates are shown, which are measured in $L^\infty $ $L^{\infty}$ -type norms in time. For the semi-discrete in time formulation, we exhibit reliable a posteriori error estimates for the error measured in the $L^\infty (L^2)$ $L^{\infty} (L^{2})$ norm with fully explicit constants; to this aim, we design a reconstruction operator into $\mathcal {C}^1$ $C^{1}$ piecewise polynomials over the time grid with optimal approximation properties in terms of the polynomial degree distribution and the time steps. Numerical examples illustrate the theoretical findings.

A priori and a posteriori error estimates of a \(\mathcal {C}^0\)-in-time method for the wave equation in second order formulation

摘要