The coefficients in the asymptotic expansion of solutions of second‐order hyperbolic problems in polygonal domains

Abstract

Solutions of boundary value problems for linear partial differential equations are known to exhibit singular behaviors near the boundary of nonsmooth domains. In this case, one is usually interested on the asymptotic behavior of the solutions near the geometric singularities. However, for both mathematical and engineering purposes, it is important to have extraction formulas for the coefficients in the asymptotic expansion. In this paper, we consider initial boundary value problems for the wave equation in two‐dimensional domains with corners and derive, by means of Fourier analysis, explicit computable expressions for the coefficients in the asymptotic expansion of the solution near the corners. The explicit structure of the singular solutions presented herein makes it fairly easy to compute numerically. Thus, the formulas can be used to construct postprocessing procedures for improving the accuracy of standard numerical techniques of approximating the solution and for improving the rates of convergence in the error estimates. The analysis presented herein would also apply to more general hyperbolic or parabolic equations with constant coefficients.

A Full Predictor-Corrector Finite Element Method for the One-Dimensional Heat Equation with Time-Dependent Singularities

“…Singular complement methods and predictor-corrector FEM (PCFEM) have been introduced to solve the problem of geometric singularities of elliptic boundary value problems (see [13]- [23]) and more recently on geometric singularities of hyperbolic problems (see [24]). The treatment of time-dependent singularities by means of PCFEM is also introduced in [25] using the method of lines.…”