The Finite Element Method for Fractional Order Viscoelasticity and the Stochastic Wave Equation

Doctoral thesis, 2009

This thesis can be considered as two parts. In the first part a hyperbolic type integro-differential equation with weakly singular kernel is considered, which is a model for dynamic fractional order viscoelasticity. In the second part, the finite element approximation of the linear stochastic wave equation is studied. The link between these two equations is that they are both treated as perturbations of the linear wave equation. Our study in the first part comprises investigating well-posedness of the model, and the analysis of the finite element approximation of the solution of the model problem. The equation, with homogeneous mixed Dirichlet and Neumann boundary conditions, is reformulated as an abstract Cauchy problem, and existence, uniqueness and regularity are verified in the context of linear semigroup theory. From a practical viewpoint, the problems with mixed homogeneous Dirichlet and non-homogeneous Neumann boundary conditions are of special importance. Therefore, the Galerkin method is used to prove existence, uniqueness and regularity of the solution of this type of problem. Then two variants of the continuous Galerkin finite element method are applied to the model problem. Stability properties of the discrete and the continuous problem are investigated. These are then used to obtain optimal order a priori estimates and global a posteriori error estimates. In a general framework, a space-time cellwise a posteriori error representation is also presented. The theory is illustrated by an example. The second part concerns the study of the semidiscrete finite element approximation of the linear stochastic wave equation with additive noise in a semigroup framework. Optimal error estimates for the deterministic problem are obtained under minimal regularity assumptions. These are used to prove strong convergence estimates for the stochastic problem. The theory presented here applies to multi-dimensional domains and correlated noise. Numerical examples illustrate the theory.