arXiv subjects

Thomas Führer

Publications and source records attributed to Thomas Führer.

At least 55 records · Page 3Linked to original sources

arXiv2018

Adaptive BEM with inexact PCG solver yields almost optimal computational costs

We consider the preconditioned conjugate gradient method (PCG) with optimal preconditioner in the frame of the boundary element method (BEM) for elliptic first-kind integral equations. Our adaptive algorithm steers the termination of PCG as well as the local mesh-refinement. Besides convergence with optimal algebraic rates, we also prove almost optimal computational complexity. In particular, we provide an additive Schwarz preconditioner which can be computed in linear complexity and which is optimal in the sense that the condition numbers of the preconditioned systems are uniformly bounded. As model problem serves the 2D or 3D Laplace operator and the associated weakly-singular integral equation with energy space $\widetilde{H}^{-1/2}(Γ)$. The main results also hold for the hyper-singular integral equation with energy space $H^{1/2}(Γ)$.

math.NA↗

arXiv2018

Fully discrete DPG methods for the Kirchhoff-Love plate bending model

We extend the analysis and discretization of the Kirchhoff-Love plate bending problem from [T. Führer, N. Heuer, A.H. Niemi, An ultraweak formulation of the Kirchhoff-Love plate bending model and DPG approximation, arXiv:1805.07835, 2018] in two aspects. First, we present a well-posed formulation and quasi-optimal DPG discretization that includes the gradient of the deflection. Second, we construct Fortin operators that prove the well-posedness and quasi-optimal convergence of lowest-order discrete schemes with approximated test functions for both formulations. Our results apply to the case of non-convex polygonal plates where shear forces can be less than $L_2$-regular. Numerical results illustrate expected convergence orders.

math.NA↗

arXiv2018

An ultraweak formulation of the Kirchhoff-Love plate bending model and DPG approximation

We develop and analyze an ultraweak variational formulation for a variant of the Kirchhoff-Love plate bending model. Based on this formulation, we introduce a discretization of the discontinuous Petrov-Galerkin type with optimal test functions (DPG). We prove well-posedness of the ultraweak formulation and quasi-optimal convergence of the DPG scheme. The variational formulation and its analysis require tools that control traces and jumps in $H^2$ (standard Sobolev space of scalar functions) and $H(\mathrm{div\,Div})$ (symmetric tensor functions with $L_2$-components whose twice iterated divergence is in $L_2$), and their dualities. These tools are developed in two and three spatial dimensions. One specific result concerns localized traces in a dense subspace of $H(\mathrm{div\,Div})$. They are essential to construct basis functions for an approximation of $H(\mathrm{div\,Div})$. To illustrate the theory we construct basis functions of the lowest order and perform numerical experiments for a smooth and a singular model solution. They confirm the expected convergence behavior of the DPG method both for uniform and adaptively refined meshes.

math.NA↗

arXiv2018

New a priori analysis of first-order system least-squares finite element methods for parabolic problems

We provide new insights into the a priori theory for a time-stepping scheme based on least-squares finite element methods for parabolic first-order systems. The elliptic part of the problem is of general reaction-convection-diffusion type. The new ingredient in the analysis is an elliptic projection operator defined via a non-symmetric bilinear form, although the main bilinear form corresponding to the least-squares functional is symmetric. This new operator allows to prove optimal error estimates in the natural norm associated to the problem and, under additional regularity assumptions, in the $L^2$ norm. Numerical experiments are presented which confirm our theoretical findings.

math.NA↗

arXiv2018

First-Order Least-Squares Method for the Obstacle Problem

We define and analyse a least-squares finite element method for a first-order reformulation of the obstacle problem. Moreover, we derive variational inequalities that are based on similar but non-symmetric bilinear forms. A priori error estimates including the case of non-conforming convex sets are given and optimal convergence rates are shown for the lowest-order case. We provide also a posteriori bounds that can be be used as error indicators in an adaptive algorithm. Numerical studies are presented.

math.NA↗

arXiv2017

Superconvergent DPG methods for second order elliptic problems

We consider DPG methods with optimal test functions and broken test spaces based on ultra-weak formulations of general second order elliptic problems. Under some assumptions on the regularity of solutions of the model problem and its adjoint, superconvergence for the scalar field variable is achieved by either increasing the polynomial degree in the corresponding approximation space by one or by a local postprocessing. We provide a uniform analysis that allows to treat different test norms. Particularly, we show that in the presence of convection only the quasi-optimal test norm leads to higher convergence rates, whereas other norms considered do not. Moreover, we also prove that our DPG method delivers the best $L^2$ approximation of the scalar field variable up to higher order terms, which is the first theoretical explanation of an observation made previously by different authors. Numerical studies that support our theoretical findings are presented.

math.NA↗

arXiv2017

Superconvergence in a DPG method for an ultra-weak formulation

In this work we study a DPG method for an ultra-weak variational formulation of a reaction-diffusion problem. We improve existing a priori convergence results by sharpening an approximation result for the numerical flux. By duality arguments we show that higher convergence rates for the scalar field variable are obtained if the polynomial order of the corresponding approximation space is increased by one. Furthermore, we introduce a simple elementwise postprocessing of the solution and prove superconvergence. Numerical experiments indicate that the obtained results are valid beyond the underlying model problem.

math.NA↗

arXiv2017

Combining the DPG method with finite elements

We propose and analyze a discretization scheme that combines the discontinuous Petrov-Galerkin and finite element methods. The underlying model problem is of general diffusion-advection-reaction type on bounded domains, with decomposition into two sub-domains. We propose a heterogeneous variational formulation that is of the ultra-weak (Petrov-Galerkin) form with broken test space in one part, and of Bubnov-Galerkin form in the other. A standard discretization with conforming approximation spaces and appropriate test spaces (optimal test functions for the ultra-weak part and standard test functions for the Bubnov-Galerkin part) gives rise to a coupled DPG-FEM scheme. We prove its well-posedness and quasi-optimal convergence. Numerical results confirm expected convergence orders.

math.NA↗

arXiv2017

A linear Uzawa-type solver for nonlinear transmission problems

We propose an Uzawa-type iteration for the Johnson-Nédélec formulation of a Laplace-type transmission problem with possible (strongly monotone) nonlinearity in the interior domain. In each step, we sequentially solve one BEM for the weakly-singular integral equation associated with the Laplace-operator and one FEM for the linear Yukawa equation. In particular, the nonlinearity is only evaluated to build the right-hand side of the Yukawa equation. We prove that the proposed method leads to linear convergence with respect to the number of Uzawa iterations. Moreover, while the current analysis of a direct FEM-BEM discretization of the Johnson-Nédélec formulation requires some restrictions on the ellipticity (resp. strong monotonicity constant) in the interior domain, our Uzawa-type solver avoids such assumptions.

math.NA↗

arXiv2016

Robust coupling of DPG and BEM for a singularly perturbed transmission problem

We consider a transmission problem consisting of a singularly perturbed reaction diffusion equation on a bounded domain and the Laplacian in the exterior, connected through standard transmission conditions. We establish a DPG scheme coupled with Galerkin boundary elements for its discretization, and prove its robustness for the field variables in so-called balanced norms. Our coupling scheme is the one from [Führer, Heuer, Karkulik: On the coupling of DPG and BEM, arXiv:1508.00630], adapted to the singularly perturbed case by using the scheme from [Heuer, Karkulik: A robust DPG method for singularly perturbed reaction diffusion problems, arXiv:1509.07560]. Essential feature of our method is that optimal test functions have to be computed only locally. We report on various numerical experiments in two dimensions.

math.NA↗

arXiv2016

On the DPG method for Signorini problems

We derive and analyze discontinuous Petrov-Galerkin methods with optimal test functions for Signorini-type problems as a prototype of a variational inequality of the first kind. We present different symmetric and non-symmetric formulations where optimal test functions are only used for the PDE part of the problem, not the boundary conditions. For the symmetric case and lowest order approximations, we provide a simple a posteriori error estimate. In a second part, we apply our technique to the singularly perturbed case of reaction dominated diffusion. Numerical results show the performance of our method and, in particular, its robustness in the singularly perturbed case.

math.NA↗

arXiv2016

A time-stepping DPG scheme for the heat equation

We introduce and analyze a discontinuous Petrov-Galerkin method with optimal test functions for the heat equation. The scheme is based on the backward Euler time stepping and uses an ultra-weak variational formulation at each time step. We prove the stability of the method for the field variables (the original unknown and its gradient weighted by the square root of the time step) and derive a Céa-type error estimate. For low-order approximation spaces this implies certain convergence orders when time steps are not too small in comparison with mesh sizes. Some numerical experiments are reported to support our theoretical results.

math.NA↗

arXiv2015

Optimal additive Schwarz methods for the $hp$-BEM: the hypersingular integral operator in 3D on locally refined meshes

We propose and analyze an overlapping Schwarz preconditioner for the $p$ and $hp$ boundary element method for the hypersingular integral equation in 3D. We consider surface triangulations consisting of triangles. The condition number is bounded uniformly in the mesh size $h$ and the polynomial order $p$. The preconditioner handles adaptively refined meshes and is based on a local multilevel preconditioner for the lowest order space. Numerical experiments on different geometries illustrate its robustness.

math.NA↗

arXiv2015

On the coupling of DPG and BEM

We develop and analyze strategies to couple the discontinuous Petrov-Galerkin method with optimal test functions to (i) least-squares boundary elements and (ii) various variants of standard Galerkin boundary elements. Essential feature of our methods is that, despite the use of boundary integral equations, optimal test functions have to be computed only locally. We apply our findings to a standard transmission problem in full space and present numerical experiments to validate our theory.

math.NA↗

arXiv2015

DPG method with optimal test functions for a fractional advection diffusion equation

We develop an ultra-weak variational formulation of a fractional advection diffusion problem in one space dimension and prove its well-posedness. Based on this formulation, we define a DPG approximation with optimal test functions and show its quasi-optimal convergence. Numerical experiments confirm expected convergence properties, for uniform and adaptively refined meshes.

math.NA↗

arXiv2015

A wirebasket preconditioner for the mortar boundary element method

We present and analyze a preconditioner of the additive Schwarz type for the mortar boundary element method. As a basic splitting, on each subdomain we separate the degrees of freedom related to its boundary from the inner degrees of freedom. The corresponding wirebasket-type space decomposition is stable up to logarithmic terms. For the blocks that correspond to the inner degrees of freedom standard preconditioners for the hypersingular integral operator on open boundaries can be used. For the boundary and interface parts as well as the Lagrangian multiplier space, simple diagonal preconditioners are optimal. Our technique applies to quasi-uniform and non-uniform meshes of shape-regular elements. Numerical experiments on triangular and quadrilateral meshes confirm theoretical bounds for condition and MINRES iteration numbers.

math.NA↗

arXiv2015

Local inverse estimates for non-local boundary integral operators

We prove local inverse-type estimates for the four non-local boundary integral operators associated with the Laplace operator on a bounded d-dimensional Lipschitz domain Omega for d >= 2 with piecewise smooth boundary. For piecewise polynomial ansatz spaces and d = 2 or 3, the inverse estimates are explicit in both the local mesh width and the approximation order. An application to efficiency estimates in a posteriori error estimation in boundary element methods is given.

math.NA↗

arXiv2014

Convergence of adaptive BEM and adaptive FEM-BEM coupling for estimators without h-weighting factor

We analyze adaptive mesh-refining algorithms in the frame of boundary element methods (BEM) and the coupling of finite elements and boundary elements (FEM-BEM). Adaptivity is driven by the two-level error estimator proposed by Ernst P. Stephan, Norbert Heuer, and coworkers in the frame of BEM and FEM-BEM or by the residual error estimator introduced by Birgit Faermann for BEM for weakly-singular integral equations. We prove that in either case the usual adaptive algorithm drives the associated error estimator to zero. Emphasis is put on the fact that the error estimators considered are {not even globally equivalent to weighted-residual error estimators for which recently convergence with quasi-optimal algebraic rates has been derived.

math.NA↗