# Adaptive local basis for elliptic problems with $\mathit{L}^{\infty}$-coefficients

Weymuth, Monika. Adaptive local basis for elliptic problems with $\mathit{L}^{\infty}$-coefficients. 2016, University of Zurich, Faculty of Science.

## Abstract

The main goal of this thesis is the development of a new finite element method for the discretization of elliptic partial differential equations in heterogeneous media. The efficient numerical modelling of such problems is of fundamental importance since they arise in many applications such as diffusion in composite materials or porous media. The challenge of modelling heterogeneous materials is that they usually have a complex structure. Often they contain complicated and/or tiny inclusions which are distributed randomly. Hence, the solution of such problems is in general non-smooth and for instance it possibly exhibits high oscillations at different length scales. Thus piecewise polynomial functions cannot resolve the essential features of the solution unless the mesh size is chosen small enough. Therefore classical polynomial-based finite element methods applied to such problems become prohibitively expensive and are not appropriate. Especially for three-dimensional problems the computational cost gets too large.
In order to overcome this problem many types of generalized finite element methods have been developed in recent years. These methods construct trial functions incorporating the physical behaviour of the solution. Thus the geometric details of the material do not have to be resolved by the mesh and the problem can be discretized by a relatively coarse mesh. The aim is the development of methods preserving the asymptotic convergence rates. This thesis is concerned with the development of a fully discrete adaptive local (AL) basis whose construction is based on a partition of unity approach. The computational domain is covered by a finite number of overlapping patches. On each patch local approximation spaces are set up by employing a discretization of the local solution operator on a fine mesh. Finally these local approximation spaces are approximated by applying an $\mathit{L}^{2}$-orthogonal projection onto low-dimensional spaces.
A complete error analysis of the method for $\mathit{L}^{\infty}$-coefficients is developed and convergence results are given. It is shown that the linear convergence property is satisfied. The error analysis is based on some new results concerning the $\mathit{W}^{1,\mathit{p}}$-regularity of the Poisson problem. These results are also presented in this thesis. Bounds for the gradient of the solution in the $\mathit{L}^{p}$-norm are derived and it is shown that they only depend on the size of the jumps in the coefficients. These regularity results can also be applied to various other problems.

## Abstract

The main goal of this thesis is the development of a new finite element method for the discretization of elliptic partial differential equations in heterogeneous media. The efficient numerical modelling of such problems is of fundamental importance since they arise in many applications such as diffusion in composite materials or porous media. The challenge of modelling heterogeneous materials is that they usually have a complex structure. Often they contain complicated and/or tiny inclusions which are distributed randomly. Hence, the solution of such problems is in general non-smooth and for instance it possibly exhibits high oscillations at different length scales. Thus piecewise polynomial functions cannot resolve the essential features of the solution unless the mesh size is chosen small enough. Therefore classical polynomial-based finite element methods applied to such problems become prohibitively expensive and are not appropriate. Especially for three-dimensional problems the computational cost gets too large.
In order to overcome this problem many types of generalized finite element methods have been developed in recent years. These methods construct trial functions incorporating the physical behaviour of the solution. Thus the geometric details of the material do not have to be resolved by the mesh and the problem can be discretized by a relatively coarse mesh. The aim is the development of methods preserving the asymptotic convergence rates. This thesis is concerned with the development of a fully discrete adaptive local (AL) basis whose construction is based on a partition of unity approach. The computational domain is covered by a finite number of overlapping patches. On each patch local approximation spaces are set up by employing a discretization of the local solution operator on a fine mesh. Finally these local approximation spaces are approximated by applying an $\mathit{L}^{2}$-orthogonal projection onto low-dimensional spaces.
A complete error analysis of the method for $\mathit{L}^{\infty}$-coefficients is developed and convergence results are given. It is shown that the linear convergence property is satisfied. The error analysis is based on some new results concerning the $\mathit{W}^{1,\mathit{p}}$-regularity of the Poisson problem. These results are also presented in this thesis. Bounds for the gradient of the solution in the $\mathit{L}^{p}$-norm are derived and it is shown that they only depend on the size of the jumps in the coefficients. These regularity results can also be applied to various other problems.