Discretization of some hyperbolic problems

Abstract

The aim of thisthesis to interest in the theoretical and numericalstudy of a differentialequation of the hyperbolic type.

In the first workweextend the Rothe method for to time discretization and finiteelementmethod for the spatial discretization of telegraphequationwithnonlocaltermassociatedwith initial conditions and boundary conditions. The main idea in thisworkis to give semi discrete and fullydiscreteschemes and extract a priori estimates and a priori errorestimates for in suitablespaces. As for the numerical aspect, the presence of non-local coefficients in the equation causes difficulties to solve a system of nonlinearequationsobtained. Therefore, wehad to addressthesedifficulties by applying a dedicatednumericalmethod to solve this type of problem, and at the end of thisworkweprovide a numericalexample to support ourtheoreticalestimates.

The purpose of the second workis to study the same non-local hyperbolicdifferentialequation by combining the $H^1$-Galerkin mixed finiteelementmethodwith the Rothe method.A priori estimates and errorestimates are derived for both semi discrete and fullydiscreteschemes in spacesthat fit thiswork and we finish ourworkwith a numericalexperimentthatprovesourtheoreticalresults.