Reduced Basis Methods for Heterogeneous Domain Decomposition Problems

Prof. Dr. Bernard Haasdonk

Immanuel Martini

01.11.2012

31.01.2017

This project aims at reduced basis (RB) approximation of heterogeneous domain decomposition problems. It comprises development of suitable methods, error analysis and efficient implementation in sophisticated software environments.

Heterogeneous domain decomposition (HDD) problems are constituted by differential equations of different types on different subdomains of the computational domain and their coupling on the interface between those subdomains. Such problems arise for example in environmental science, biology or food technologies. A classical example is the modelling of groundwater flow, where the distinction of free flow and flow through permeable material leads to a coupled system of the Stokes equation and the porous media equation.

To get accurate simulation results high dimensional discretization schemes have to be used, which result in very high computational times. This gets extremely costly if many simulations have to be performed, whilst the underlying aggregate systems are of the same type but differ in few parameter values. In such scenarios the reduced basis (RB) method is a very powerful model reduction method that enables online simulations with drastically reduced computational times.

We aim to efficiently treat HDD problems by the RB method. In doing so we build upon recent developments including the static condensation RB element method, the Dirichlet-Neumann RB method and the reduced basis-domain decomposition-finite element method (RDF), all of which treat homogeneous domain decomposition problems. A critical point is the number of global high dimensional solves our method relies on. The decomposed nature of the problems allows to construct reduced bases on the subdomains with potentially few global solution information and so gives an advantage over treatment of the global aggregate system by means of the standard RB method. At the same time the global accuracy must be reliable and shall be indicated by rapidly computable a-posteriori error bounds.

The work also comprises implementation of the methods and numerical validation of the results in RBmatlab, a MATLAB software package for RB methods, and in DUNE, a C++ toolbox for solving partial differential equations.