Convergence Analysis for Multi-level Spectral Deferred Corrections

The spectral deferred correction (SDC) method is an iterative solver for ordinary differential equations (ODEs). It can be interpreted as a preconditioned Picard iteration for the collocation problem. The convergence of this method is well-known, it gains one order per iteration up to the order of the quadrature method of the collocation problem. This appealing feature enables an easy creation of flexible, high-order accurate methods for ODEs. A variation of SDC are multi-level spectral deferred corrections (MLSDC). Here, sweeps are performed on a hierarchy of levels and an FAS correction term, as in nonlinear multigrid methods, couples solutions on different levels. While there are several numerical examples which show its correctness and efficiency, a theoretical convergence proof is still missing. This thesis addresses this issue. A proof of the convergence of MLSDC, including the determination of the convergence order, will be outlined and the results of the theoretical analysis will be numerically demonstrated. It turns out that there are restrictions for the superiority of this method over SDC regarding the convergence order