**Eldorado - Repository of the TU Dortmund**

Resources for and from Research, Teaching and Studying

### Recent Submissions

New results on the theoretical solvability of nonlinear algebraic flux correction (AFC) problems are presented and a Newton-like solution technique exploiting an efficient computation of the Jacobian is introduced. The AFC methodology is a rather new and unconventional approach to algebraically stabilize finite element discretizations of convection-dominated transport problems in a bound-preserving manner. Besides investigations concerning the theoretical solvability, the development of effic...

The aim of this paper is to describe a new, fast and robust solver for 3D flow problems which are described by the incompressible Navier-Stokes equations. The correspondig simulations are done by a monolithic 3D flow solver, i.e. velocity and pressure are solved at the same time. During these simulations the convective part is linearized using two different methods: Fixpoint method and Newton method. The Fixpoint method is working in a quite robust way, but it has a slow convergence dependi...

This paper is concerned with a priori error estimates for the local incremental minimization scheme, which is an implicit time discretization method for the approximation of rate-independent systems with non-convex energies. We first show by means of a counterexample that one cannot expect global convergence of the scheme without any further assumptions on the energy. For the class of uniformly convex energies, we derive error estimates of optimal order, provided that the Lipschitz const...

Using the theoretical framework of algebraic flux correction and invariant domain preserving schemes, we introduce a monolithic approach to convex limiting in continuous finite element schemes for linear advection equations, nonlinear scalar conservation laws, and hyperbolic systems. In contrast to fluxcorrected transport (FCT) algorithms that apply limited antidiffusive corrections to bound-preserving low-order solutions, our new limiting strategy exploits the fact that these solutions...

We consider finite element solutions to quadratic optimization problems, where the state depends on the control via a well-posed linear partial differential equation. Exploiting the structure of a suitably reduced optimality system, we prove that the combined error in the state and adjoint state of the variational discretization is bounded by the best approximation error in the underlying discrete spaces. The constant in this bound depends on the inverse square-root of the Tikhonov regulariza...

We investigate the problem of optimal transport in the so-called Kantorovich form, i.e. given two Radon measures on two compact sets, we seek an optimal transport plan which is another Radon measure on the product of the sets that has these two measures as marginals and minimizes a certain cost function. We consider quadratic regularization of the problem, which forces the optimal transport plan to be a square integrable function rather than a Radon measure. We derive the dual problem and sho...

In this work, we introduce a new residual distribution (RD) framework for the design of matrix-free bound-preserving finite element schemes. As a starting point, we consider continuous and discontinuous Galerkin discretizations of the linear advection equation. To construct the corresponding local extremum diminishing (LED) approximation, we perform mass lumping and redistribute the element residuals in a manner which guarantees the LED property. The hierarchical correction procedure for high...

This paper is concerned with a space-time discretization of a rate-independent evolution governed by a non-smooth dissipation and a non-convex energy functional. For the time discretization, we apply the local minimization scheme introduced in [EM06], which is known to resolve time discontinuities, which may show up due to the non-convex energy. The spatial discretization is performed by classical linear finite elements. We show that accumulation points of the sequence of discrete solutions f...

In a posteriori error analysis, the relationship between error and estimator is usually spoiled by so-called oscillation terms, which cannot be bounded by the error. In order to remedy, we devise a new approach where the oscillation has the following two properties. First, it is dominated by the error, irrespective of mesh fineness and the regularity of data and the exact solution. Second, it captures in terms of data the part of the residual that, in general, cannot be quantified with finite...

The article at hand focuses on finite element discretizations, where the continuous and the discrete formulations differ. We introduce a general approach based on the dual weighted residual method for estimating on the one hand the discretization error in a user specified quantity of interest and on the other hand the discrete model error induced by using different discrete techniques. Here, the usual error identities are obtained plus some additional terms. Furthermore, the numerical approxi...

This paper is focused on efficient Monte Carlo simulations of Brownian diffusion effects in particle-based numerical methods for solving transport equations on a sphere (or a circle). Using the heat equation as a model problem, random walks are designed to emulate the action of the Laplace-Beltrami operator without evolving or reconstructing the probability density function. The intensity of perturbations is fitted to the value of the rotary diffusion coefficient in the deterministic mo...

We approximate the solution of the stationary Stokes equations with various conforming and nonconforming inf-sup stable pairs of finite element spaces on simplicial meshes. Based on each pair, we design a discretization that is quasi-optimal and pressure robust, in the sense that the velocity H^1-error is proportional to the best H^1-error to the analytical velocity. This shows that such a property can be achieved without using conforming and divergence-free pairs. We bound also the pressure ...

We develop a basic convergence analysis for an adaptive C0IPG method for the Biharmonic problem which provides convergence without rates for all practically relevant marking strategies and all penalty parameters assuring coercivity of the method. The analysis hinges on embedding properties of (broken) Sobolev and BV spaces, and the construction of a suitable limit space. In contrast to the convergence result of adaptive discontinuous Galerkin methods for elliptic PDEs, by Kreuzer and Georgoul...