Reduced basis for nonlinear diffusion equations

• Die reduzierte Basis für nichtlineare Diffusionsgleichungen

Rasty, Mohammad; Grepl, Martin Alexander (Thesis advisor); Herty, Michael Matthias (Thesis advisor)

Aachen (2016)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen, 2016

Abstract

The Reduced Basis Method (RBM) is a model order reduction technique for solving parametric partial differential equations (PDEs) by reducing the dimension of the underlying problem. The essence of the RBM is to divide the computational process into two stages: an offline and an online stage. In the offline stage, which is performed only once, the reduced basis is generated and required information needed in the online stage is computed and saved. In the online stage, the parameter dependent PDE is solved very efficiently by utilizing data provided from the offline stage.In this thesis, the RBM is extended to treat nonlinear diffusion equations. We first consider an elliptic and parabolic quadratically nonlinear diffusion equation. In the elliptic case, the reduced basis approximation is based on a Galerkin projection and the Brezzi-Rappaz-Raviart (BRR) framework is used to derive rigorous a posteriori error bounds. We subsequently extend these results to the parabolic case by combining the BRR framework with the space-time method. We show that the reduced basis approximation and the associated a posteriori bounds can be computed using an efficient offline-online computational framework, both for the elliptic and parabolic case.In the second part of this thesis, we focus on higher order nonlinear diffusion equations. Higher order nonlinearities pose an additional challenge for model order reduction methods, since the dimensional reduction often does not result in a computational gain. The reason lies in the often expensive evaluation of the nonlinearity, i.e. there is no complete decoupling of the online stage from the underlying high-dimensional problem. One possible approach to solve this problem is the Empirical Interpolation Method (EIM), which allows to approximate the nonlinearity by an affine representation of previously computed basis functions using interpolation. We again consider both the steady-state and time-dependent case, and develop reduced basis approximations and associated {\it a posteriori} error bounds for higher order nonlinear diffusion equations. We remark that although the EIM provides an efficient offline-online computational procedure, the error estimators evaluated using the EIM might not be rigorous.