A monolithic operator-adaptive Newton-Multigrid solver for Navier-Stokes Equations in 3D See

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...

A-priori error analysis of local incremental minimization schemes for rate-independent evolutions See

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...

Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws See

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...

Limiting and divergence cleaning for continuous finite element discretizations of the MHD equations See

Radiale Basisfunktionen See

Musik und Mathematik See

PP-Löser für die Navier-Stokes Gleichung See

A stabilized DG cut cell method for discretizing the linear transport equation See

Lineare Inverse Probleme See

Quasi-best approximation in optimization with PDE constraints See

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...

Regularization for optimal control problems associated to nonlinear evolution equations See

Quadratically regularized optimal transport See

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...

Matrix-free subcell residual distribution for Bernstein finite elements: Low-order schemes and FCT See

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...

Finite Element Discretization of Local Minimization Schemes for Rate-Independent Evolutions See

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...

Oscillation in a posteriori error estimation See

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...

Goal oriented a posteriori error estimators for problems with modified discrete formulations based on the dual weighted residual method See

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...

Random walk methods for Monte Carlo simulations of Brownian diffusion on a sphere See

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...

Quasi-optimal and pressure robust discretizations of the Stokes equations by new augmented Lagrangian Formulations See

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 ...

Convergence of adaptive C0-interior penalty Galerkin method for the biharmonic problem See

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...