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.
continuous Galerkin method
fractional order viscoelasticity
a posteriori error estimate
weakly singular kernel
a priori error estimate
stochastic wave equation
finite element method
room Pascal, Department of Mathematical Sciences, Chalmers Tvargata 3, Chalmers University
Opponent: Dr. Omar Lakkis, Department of Mathematics, University of Sussex, England.