Analysis of parallel algorithms for time-dependent partial differential equations

For time dependent partial differential equations (PDEs) there are two established methods to obtain algorithms which can be run in parallel: If the time is discretized, one obtains a sequence of time independent problems which can be solved in parallel using domain decomposition. On the other hand if the space is discretized, one obtains a large system of ordinary differential equations which can be solved in parallel using waveform relaxation.; Both approaches rely heavily on the theory developed specifically for the problems obtained after the discretization. Convergence results are thus limited to the results which are known for domain decomposition and waveform relaxation.; I take a different approach in this thesis by analyzing parallel algorithms without discretizing the PDE. This has the main advantage that one can work at the continuous level for which the classical tools of analysis are well suited. Furthermore the properties of the differential operators are not hidden in a discretization and can therefore be fully exploited in the design of the parallel algorithm.; In the first part of my thesis I analyze the convergence of the overlapping Schwarz algorithm for parabolic and hyperbolic equations. In the parabolic case I show linear convergence of the algorithm on unbounded time intervals and superlinear convergence on bounded time intervals. In the hyperbolic case I show convergence in a finite number of steps on bounded time intervals and divergence on unbounded time intervals. I show furthermore how these results generalize to equations with variable coefficients and nonlinear terms.; In the second part I investigate a more general subspace decomposition for semilinear parabolic evolution equations in a Hilbert space. I consider subspaces of the Hilbert space and the evolution of the solutions projected into those subspaces while fixing the solution in the remaining subspaces. I prove linear convergence of this iteration on unbounded time intervals under a strong Lipschitz condition on the nonlinear term and superlinear convergence on bounded time intervals under a mild Lipschitz condition. As an example I consider subspaces defined by a frequency decomposition in eigenfrequencies of the linear operator.