Stefan A. Funken

Ulm University | UULM · Institute of Numerical Mathematics

The edge corrosion simulation of Kapfer et al. is used to extract time‐dependent delamination widths of metal sheet edges with different spatial orientations, encoded by edge classes and angular displacements with respect to the ground. The corrosion behavior of the edges of a body‐in‐white part is examined with respect to the spatial orientation a...

Adaptive meshing is a fundamental component of adaptive finite element methods. This includes refining and coarsening meshes locally. In this work, we are concerned with the red-green-blue refinement strategy in two dimensions and its counterpart-coarsening. In general, coarsening algorithms are mostly based on an explicitly given refinement histor...

The $L^2$-orthogonal projection $ {\varPi }_h:L^2(\varOmega )\rightarrow {\mathbb{V}}_h$ onto a finite element (FE) space $ {\mathbb{V}}_h$ is called $H^1$-stable iff $ \|{ {\varPi }_h u}\|_{H^{1}}(\varOmega )\leq C\|{u}\|_{H^{1}}(\varOmega ) $ for any $u\in H^1(\varOmega )$ with a positive constant $C\neq C(h)$ independent of the mesh size $h>0$....

Scaling finite element method (FEM) based corrosion simulations to whole body‐in‐white structures lead to extremely high computational costs. As corrosion only appears in corrosive critical areas, the FEM is restricted to these. The objective is a semantic segmentation of corrosive critical designs, which are part edges and flanges in body‐in‐white...

Adaptive meshing includes local refinement as well as coarsening of meshes. Typically, coarsening algorithms are based on an explicit refinement history. In this work, we deal with local coarsening algorithms that build on the refinement strategies for triangular and quadrilateral meshes implemented in the ameshref package (Funken and Schmidt 2018,...

The $L^2$-orthogonal projection $\Pi_h:L^2(\Omega)\rightarrow\mathbb{V}_h$ onto a finite element (FE) space $\mathbb{V}_h$ is called $H^1$-stable iff $\|\nabla\Pi_h u\|_{L^2(\Omega)}\leq C\|u\|_{H^1(\Omega)}$, for any $u\in H^1(\Omega)$ with a positive constant $C\neq C(h)$ independent of the mesh size $h>0$. In this work, we discuss local criteria...

Adaptive meshing is a fundamental component of adaptive finite element methods. This includes refining and coarsening meshes locally. In this work, we are concerned with the red-green-blue refinement strategy and its counterpart - coarsening. In general, coarsening algorithms are mostly based on an explicitly given refinement history. In this work,...

This paper deals with the efficient implementation of various adaptive mesh refinements in two dimensions in Matlab. We give insights into different adaptive mesh refinement strategies allowing triangular and quadrilateral grids with and without hanging nodes. Throughout, the focus is on an efficient implementation by utilization of reasonable data...

This paper presents a Matlab-Toolbox named ameshref that provides an efficient implementation of various adaptive mesh refinement strategies allowing triangular and quadrilateral grids with and without hanging nodes. For selected methods, we give an insight into the strategy itself and the core ideas for an efficient realization. This is achieved b...

The transport of intensity equation (TIE) provides a very straight forward way to computationally reconstruct wavefronts from measurements of the intensity and the derivative of this intensity along the optical axis of the system. However, solving the TIE requires knowledge of boundary conditions which cannot easily be obtained experimentally. The...

We discuss the accurate and efficient implementation of hp -BEM for the Laplace operator in two dimensions. Using Legendre polynomials and their antiderivatives as local bases for the discrete ansatz spaces, we are able to reduce both the evaluation of potentials and the computation of Galerkin entries to the evaluation of basic integrals. For the...

A posteriori error estimation is an important tool for reliable and efficient Galerkin boundary element computations. For hypersingular integral equations in 2D with a positive-order Sobolev space, we analyse the mathematical relation between the (h − h/2)-error estimator from [S. Ferraz-Leite and D. Praetorius, Simple a posteriori error estimators...

We provide a MATLAB package p1afem for an adaptive P1-finite element method (AFEM). This includes functions for the assembly of the data, different error estimators, and an indicator-based adaptive meshrefining algorithm. Throughout, the focus is on an efficient realization by use of MATLAB built-in functions and vectorization. Numerical experim...

Recent developments of solid electrolytes, especially lithium ion conductors, led to all solid state batteries for various applications. In addition, mathematical models sprout for different electrode materials and battery types, but are missing for solid electrolyte cells. We present a mathematical model for ion flux in solid electrolytes, based o...

We present a new model for all solid-state lithium ion batteries that takes into account detailed aspects of ion transport in solid solutions of crystalline metal oxides. More precisely, our model describes transient lithium ion flux through a solid electrolyte, the solid–solid interfaces and an intercalation electrode. The diffuse part of the dou-...

A posteriori error estimation is an important tool for reliable and efficient Galerkin boundary element computations. We analyze the mathematical relation between the h-h/2-error estimator from [S. Ferraz-Leite, D. Praetorius, Simple a posteriori error estimators for the h-version of the boundary element method, Computing 83 (2008) 135–162], the tw...

The purpose of this paper is to present optimal preconditioned iterative methods to solve indefinite linear systems of equations arising from symmetric coupling of finite elements and boundary elements. This is a block-diagonal preconditioner together with a conjugate residual method and a preconditioned inner–outer iteration. We prove the efficien...

The Boolean model of Wiener sausages is a random closed set that can be thought of as a random collection of parallel neighborhoods of independent Wiener paths in space. It describes e.g. the target detection area of a network of sensors moving according to the Brownian dynamics whose initial locations are chosen in the medium at random. In the pap...

The magnetization of a ferromagnetic sample solves a non-convex variational problem, where its relaxation by convexifying the energy density resolves relevant macroscopic information. The numerical analysis of the relaxed model has to deal with a constrained convex but degenerated, nonlocal energy functional in mixed formulation for magnetic potent...

A short Matlab implementation for P 1 and Q 1 finite elements (FE) is provided for the numerical solution of 2d and 3d problems in linear elasticity with mixed boundary conditions. Any adaptation from the simple model examples provided to more complex problems can easily be performed with the given documentation. Numerical examples with postprocess...

In the third part of our investigations on averaging techniques for a posteriori error control in elasticity we focus on nonconforming finite elements in two dimensions. Kouhia and Stenberg [Comput. Methods Appl. Mech. Engrg. 124 (1995) 195] established robust a priori error estimates for a Galerkin-discretisation where the first component of the d...

In the second part of our investigation on a posteriori error estimates and a posteriori error control in finite element analysis in elasticity, we focus on robust a posteriori error bounds. First we establish a residual-based a posteriori error estimate which is reliable and efficient up to higher-order terms and λ-independent multiplicative const...

This paper contains a systematic study of numerical approximations for solving the exact kernel form of Pocklington's integro-differential equation for the current induced on a thin wire by an incident time-harmonic electromagnetic field. We consider various Galerkin (h, p, hp, and adaptive h) and collocation schemes and show that a sensible hp ref...

Averaging techniques are popular tools in adaptive finite element methods for the numerical treatment of second-order partial differential equations since they provide efficient a posteriori error estimates by a simple postprocessing. In this paper, the reliability of any averaging estimator is shown for low order finite element methods in elastici...

This paper contains a systematic study of numerical approximations for solving the exact kernel form of Pocklington's integro-dierential equation for the current induced on a thin wire by an incident time harmonic electromagnetic eld. We consider various Galerkin (h, p, hp and adaptive h) and collocation schemes, and show that a sensible hp renemen...

Mixed finite element methods (mfem) are of particular interest e.g. in elasticity where locking phenomena can be circumvented. But they are restricted to bounded domains. This is contrary to the boundary element methods (bem) which can be applied to the most important linear and homogeneous partial differential equations with constant coefficients...

. The magnetization state of a ferromagnetic body is given as the solution of a non-convex variational problem. A relaxation of this model by convexifying the energy density resolves essential macroscopic information that applied physicists and engineers are after. The numerical analysis of the relaxed model faces a nonlinearly constrained convex b...

If the first task in numerical analysis is the calculation of an approximate solution, the second is to provide a guaranteed error bound and is often of equal importance. The standard approaches in the a posteriori error analysis of finite element methods suppose that the exact solution has a certain regularity or the numerical scheme enjoys some s...

. Computable a posteriori error bounds and related adaptive mesh-refining algorithms are provided for the numerical treatment of monotone stationary flow problems with a quite general class of conforming and nonconforming finite element methods. A refined residual-based error estimate generalises the works of Verfurth, Dari, Duran & Padra, Bao & Ba...

A short Matlab implementation for P 1-x 1 finite elements on triangles and parallelograms is provided for the numerical solution of elliptic problems with mixed boundary conditions on unstructured grids. According to the shortness of the program and the given documentation, any adaptation from simple model examples to more complex problems can easi...

Nonconforming finite element methods are sometimes considered as a variational crime and so we may regard its coupling with boundary element methods. In this paper, the symmetric coupling of nonconforming finite elements and boundary elements is established and a priori error estimates are shown. The coupling involves a further continuous layer on...

The coupling of nonconforming finite element and boundary element methods was established in Part I of this paper, where quasi-optimal a priori error estimates are provided. In the second part, we establish sharp a posteriori error estimates and so justify adaptive mesh-refining algorithms for the efficient numerical treatment of transmission probl...

Some approaches in the a posteriori error analysis of finite element methods (FEM) are based on the regularity of the exact solution or on a saturation property of the numerical scheme. For coarse meshes those asymptotic arguments are difficult to recast into rigorous error bounds. Here, we will provide reliable computable error bounds which are ef...

. The symmetric coupling of mixed finite element and boundary element methods is analysed for a model interface problem with the Laplacian. The coupling involves a further continuous ansatz function on the interface to link the discontinuous displacement field to the necessarily continuous boundary ansatz function. Quasi-optimal a priori error esti...

Mixed finite element methods such as PEERS or the BDMS methods are designed to avoid locking for nearly incompressible materials in plane elasticity. In this paper, we establish a robust adaptive mesh-refining algorithm that is rigorously based on a reliable and efficient a posteriori error estimate. Numerical evidence is provided for the -independ...

. A short Matlab implementation for P 1 -Q 1 finite elements on triangles and parallelograms is provided for the numerical solution of elliptic problems with mixed boundary conditions on unstructured grids. According to the shortness of the programme and a given documentation, any adaptation from simple model examples to more complex problems can e...

. This paper is concerned with the coupling of non-conform finite element and boundary element methods in continuation of Part I (C. Carstensen, S.A. Funken: Coupling of non-conform finite elements and boundary elements I: a priori estimates), where we recast the interface model problem, introduced a coupling scheme and proved a priori error estima...

This paper presents fast iterative solvers for coupled non-linear Finite Element and Boundary Element problems using a damped inexact Newton method á la Axellson and Kaporin. This method converges globally even if the second Gateaux-dervitative does not exist. The used solvers for the linear saddle point problems occuring in the modified Newton alg...

This paper concerns the combination of the finite element method (FEM) and the boundary element method (BEM) using the symmetric coupling. As a model problem in two dimensions we consider the Hencky material (a certain nonlinear elastic material) in a bounded domain with Navier–Lamé differential equation in the unbounded complementary domain. Using...

This paper deals with a general framework for a posteriori error estimates in boundary element methods which is specified for three examples, namely Symm's integral equation, an integral equation with a hypersingular operator, and a boundary integral equation for a transmission problem. Based on these estimates, an analog of Eriksson and Johnson's...

: This paper presents a posteriori error estimates for the hp--version of the boundary element method. We discuss two first kind integral operator equations, namely Symm's integral equation and the integral equation with a hypersingular operator. The computable upper error bounds indicate an algorithm for the automatic hp--adaptive mesh--refinement...

The purpose of this paper is to present a nearly optimal preconditioned iterative method to solve indefinite linear systems of equations arising from h-adaptive procedures for the symmetric coupling of Finite Elements and Boundary Elements. This solver is nearly optimal in the sense, that its convergence rate grows only logarithmically with the num...

In this paper we discuss the BPX preconditioner for the single layer potential operator. We find that the extrem eigenvalues of the preconditioner applied to the single layer potential operator are bounded independent of the number of unknowns. A description of an efficient implementation of the BPX algorithm is given. Subject Classifications: AMS(...

The purpose of this paper is to present two optimal preconditioned iterative methods to solve indefinite linear systems of equations arising from symmetric coupling of finite elements and boundary elements [8]. These are a block-diagonal preconditioner together with a conjugate residual method (PCR) and a preconditioned inner-outer iteration (PIO)....

In our talk, we propose an adaptive mesh-refining strategy fo r the cell-centered FVM based on some a posteriori error control for the quantityk∇ T (u − Iuh)k L2 . Here, uh ∈ P 0 (T ) denotes the FVM approximation ofu and I is a certain interpolation opera- tor. As model example serves the Laplace equation with mixed boundary conditions, where our...

We consider the reliability of averaging techniques for low-order finite element methods. Emphasis is on unstructured grids using conforming, nonconforming and partly nonconforming methods. Theoretical and numerical evidence supports that the reliability depends on the smoothness of given right-hand sides, and is independent of the structure of sha...

Kiel, Universiẗat, Habil.-Schr., 2002.