Wavelet compressed, modified Hilbert transform in the space-time discretization of the heat equation

On a finite time interval $(0,T)$, we consider the multiresolution Galerkin discretization of a modified Hilbert transform $\mathcal H_T$ which arises in the space-time Galerkin discretization of the linear diffusion equation. To this end, we design spline-wavelet systems in $(0,T)$ consisting of piecewise polynomials of degree $\geq 1$ with sufficiently many vanishing moments which constitute Riesz bases in the Sobolev spaces $H^{s}_{0,}(0,T)$ and $H^{s}_{,0}(0,T)$. These bases provide multilevel splittings of the temporal discretization spaces into "increment" or "detail" spaces of direct sum type. Via algebraic tensor-products of these temporal multilevel discretizations with standard, hierarchic finite element spaces in the spatial domain (with standard Lagrangian FE bases), sparse space-time tensor-product spaces are obtained, which afford a substantial reduction in the number of the degrees of freedom as compared to time-marching discretizations. In addition, temporal spline-wavelet bases allow to compress certain nonlocal integrodifferential operators which appear in stable space-time variational formulations of initial-boundary value problems, such as the heat equation and the acoustic wave equation. An efficient preconditioner is proposed that affords linear complexity solves of the linear system of equations which results from the sparse space-time Galerkin discretization.Comment: 32 page

Space-time goal-oriented error control and adaptivity for discretizations and reduced order modeling of multiphysics problems

In this thesis, we investigate the use of adaptive methods for the efficient solution of linear multiphysics problems and nonlinear coupled problems. The main ingredients are a posteriori error estimates based on the dual-weighted residual method. By solving an auxiliary adjoint problem, these error estimates can be used to compute local error indicators for spatial and temporal refinements, which can be used for adaptive spatial and temporal meshes for e.g. the Navier-Stokes equations. For interface- and volume-coupled problems, we present a further extension of temporal adaptivity by using different temporal meshes for each subproblem while still being able to assemble the linear system in a monolithic fashion. Since multiphysics problems, like poroelasticity, are expensive to solve for fine discretizations with millions of degrees of freedom, we present a novel online-adaptive model order reduction method called MORe DWR (Model Order Reduction with Dual-Weighted Residual error estimates), which merges classical proper orthogonal decomposition based model order reduction with a posteriori error estimates. Thus, we can avoid the costly offline phase of classical model order reduction methods and still achieve high accuracy by enriching the reduced basis on-the-fly in the online phase when the error estimators exceed a given tolerance.German Research Foundation (DFG)/International Research Training Group 2657/Grant Number 43308229/E