Space-Time Reduced Basis Method for Solving Parameterized Heat Equations

Abstract

In this thesis, we present a reduced basis space-time finite element method for solving the parameterized initial boundary value heat equation. After showing the unique solvability of the underlying problem, we discretize it by using a full space-time Petrov-Galerkin finite element discretization. The space-time finite element approach uses a variational formulation in both time and space and allows for unstructured finite elements without any tensor structure. This makes the mesh more general and due to the variational treatment of space and time many solution techniques from elliptic equations can be taken over by slight modifications. In this way, we also give a parabolic version of the reduced basis method to solve the parameterized heat equation for varying parameters extremely fast (''real-time'') and very often (''multi-query''). Due to the speedup of the computational time with the reduced model the question arises how well the reduced solution approximates the finite element solution. Therefore, an a posteriori error estimator is derived to guarantee the accuracy of the reduced model. This error estimator will then be of crucial importance in the greedy algorithm for generating a reduced basis. In the numerical experiments we confirm the derived a priori error estimator between a known solution and its finite element approximation as well as the derived a posteriori error estimator between the finite element solution and the reduced basis approximation. Furthermore, we investigate the accuracy and computational complexity of the reduced basis method with a greedy generated basis. These tests show that the accuracy between reduced and finite element approximation can be chosen sufficiently well and at the same time the dimension of the reduced system is indeed much smaller than the dimension of the finite element system, which allows a faster evaluation of the solution.