Carsten Gräser

Christian Engwer, Carsten Gräser, Steffen Müthing et al.

The dune-functions Dune module provides interfaces for functions and function space bases. It forms one abstraction level above grids, shape functions, and linear algebra, and provides infrastructure for full discretization frameworks like dune-pdelab and dune-fem. This document describes the function space bases provided by dune-functions. These are based on an abstract description of bases for product spaces as trees of simpler bases. From this description, many different numberings of degrees of freedom by multi-indices can be derived in a natural way. We describe the abstract concepts, document the programmer interface, and give a complete example program that solves the stationary Stokes equation using Taylor-Hood elements.

Carsten Gräser, Oliver Sander

The Truncated Nonsmooth Newton Multigrid (TNNMG) method is a robust and efficient solution method for a wide range of block-separable convex minimization problems, typically stemming from discretizations of nonlinear and nonsmooth partial differential equations. This paper proves global convergence of the method under weak conditions both on the objective functional, and on the local inexact subproblem solvers that are part of the method. It also discusses a range of algorithmic choices that allows to customize the algorithm for many specific problems. Numerical examples are deliberately omitted, because many such examples have already been published elsewhere.

Carsten Gräser, Max Kahnt, Ralf Kornhuber

We consider a non-isothermal multi-phase field model. We subsequently discretize implicitly in time and with linear finite elements. The arising algebraic problem is formulated in two variables where one is the multi-phase field, and the other contains the inverse temperature field. We solve this saddle point problem numerically by a non-smooth Schur-Newton approach using truncated non-smooth Newton multigrid methods. An application in grain growth as occurring in liquid phase crystallization of silicon is considered.

Carsten Gräser, Tobias Kies

This paper is concerned with minimization of a fourth-order linearized Canham-Helfrich energy subject to Dirichlet boundary conditions on curves inside the domain. Such problems arise in the modeling of the mechanical interaction of biomembranes with embedded particles. There, the curve conditions result from the imposed particle--membrane coupling. We prove almost-$H^{\frac{5}{2}}$ regularity of the solution and then consider two possible penalty formulations. For the combination of these penalty formulations with a Bogner-Fox-Schmit finite element discretization we prove discretization error estimates which are optimal in view of the solution's reduced regularity. The error estimates are based on a general estimate for linear penalty problems in Hilbert spaces. Finally, we illustrate the theoretical results by numerical computations. An important feature of the presented discretization is that it does not require to resolve the particle boundary. This is crucial in order to avoid re-meshing if the presented problem arises as subproblem in a model where particles are allowed to move or rotate.

Carsten Gräser, Prem Anand Alathur Srinivasan

In this work we consider the regularization of a supervised learning problem by partial differential equations (PDEs) and derive error bounds for the obtained approximation in terms of a PDE error term and a data error term. Assuming that the target function satisfies an unknown PDE, the PDE error term quantifies how well this PDE is approximated by the auxiliary PDE used for regularization. It is shown that this error term decreases if more data is provided. The data error term quantifies the accuracy of the given data. Furthermore, the PDE-regularized learning problem is discretized by generalized Galerkin discretizations solving the associated minimization problem in subsets of the infinite dimensional functions space, which are not necessarily subspaces. For such discretizations an error bound in terms of the PDE error, the data error, and a best approximation error is derived.