Finite Element Analysis for Strongly Damped Wave Equation

Abstract

Finite element methods (FEMs) for the strongly damped wave equation with semi- and fully discrete solutions are presented in this paper. First, a system of first-order equations with respect to time was created from the strongly damped wave equation. We established the solution spaces for the numerical solution by identifying the stability bounds of the solution. Convergence to the continuous solution is demonstrated as part of the analysis. Additionally, a practicable algorithm is suggested to effectively solve the fully-discrete formulation at each time step. The calculations are performed in one, two, and three spatial dimensions, demonstrating
the efficacy of the suggested methodology. The special characteristics of the study include the thorough transformation of the second-order equation into a first-order system in time, the establishment of weak solutions, the proof and derivation of stability bounds, and a convergence analysis of the proposed FEM schemes.

Keywords: damped wave equation; semi-discrete; fully-discrete; stability bounds; existence.

2010 Mathematics Subject Classification. 65M60; 65M12; 35D30