Higher order triangular finite elements with mass lumping for the wave equation.

*(English)*Zbl 1019.65077The purpose of this article is to construct classes of \(H^1\) conforming triangular polynomial finite element space of order two and three for solving the wave equation in arbitrary two-dimensional domains. The goal is to use suitably higher order quadrature rules such that the resulting mass matrix (the matrix of all inner products of polynomial elements) becomes positive definite and diagonal.

The authors demonstrate that this is a nontrivial task and achieve this by an explicit construction in slightly larger polynomial spaces. Convergence of the method is analyzed by applying the Laplace transform to the semidiscretized wave equation, which permits the use of well-known techniques for elliptic problems. Modified leapfrog schemes are proposed for the fully discretized problem in order to obtain higher order accuracy also in the time domain. Numerical examples are also given, including one with complex geometry (diffraction off a drop shaped obstacle).

The authors demonstrate that this is a nontrivial task and achieve this by an explicit construction in slightly larger polynomial spaces. Convergence of the method is analyzed by applying the Laplace transform to the semidiscretized wave equation, which permits the use of well-known techniques for elliptic problems. Modified leapfrog schemes are proposed for the fully discretized problem in order to obtain higher order accuracy also in the time domain. Numerical examples are also given, including one with complex geometry (diffraction off a drop shaped obstacle).

Reviewer: Hans Engler (Bonn)

##### MSC:

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods 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 |

35L05 | Wave equation |