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

Zhaonan Dong, Lorenzo Mascotto, Zuodong Wang

Abstract

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 and piecewise continuous polynomial in time test and trial spaces, 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 $L^\infty$-type norms in time. For the semi-discrete in time formulation, we exhibit reliable a posteriori error estimates for the error measured in the $L^\infty(L^2)$ norm with fully explicit constants; to this aim, we design a reconstruction operator into $\mathcal C^1$ 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

Abstract

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 and piecewise continuous polynomial in time test and trial spaces, 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 -type norms in time. For the semi-discrete in time formulation, we exhibit reliable a posteriori error estimates for the error measured in the norm with fully explicit constants; to this aim, we design a reconstruction operator into 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.

Paper Structure

This paper contains 23 sections, 15 theorems, 167 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. A priori error analysis
  3. Stability estimates
  4. An integrated Thomée-type operator
  5. Abstract error analysis
  6. error estimates
  7. A posteriori error estimates for the semi-discrete in time method
  8. Data assumptions
  9. A reconstruction operator
  10. An error estimator for the semi-discrete in time method
  11. Numerical experiments
  12. Uniform refinements
  13. Uniform refinements: test case 1
  14. Uniform refinements: test case 2
  15. Simultaneous space--time uniform refinements: test case 3
  16. ...and 8 more sections

Key Result

Theorem 2.1

Let $U_{h}$ be the solution to Walkington:method and $f$ be the source term in strong-problem. The following stability estimate holds trueThe norms of $U_{h}'$ increase cubically in $p$ with respect to to norm of $f$, and with rate $p^\frac{3}{2}$ with respect to the norm of the initial conditions.:

Figures (10)

  • Figure 1: Exact solution as in \ref{['test-case:1']}, uniform $\boldsymbol \tau$-refinement.
  • Figure 2: Exact solution as in \ref{['test-case:1']}, uniform $p_n^t$-refinement.
  • Figure 3: Exact solution as in \ref{['test-case:2']}, uniform $\boldsymbol \tau$-refinement.
  • Figure 4: Exact solution as in \ref{['test-case:2']}, $p_n^t$-refinement.
  • Figure 5: Exact solution as in \ref{['test-case:3']}, $\boldsymbol \tau$-refinement.
  • ...and 5 more figures

Theorems & Definitions (34)

  • Theorem 2.1
  • proof
  • Remark 1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 24 more