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Alex Kaltenbach

Publications and source records attributed to Alex Kaltenbach.

At least 37 records · Page 2Linked to original sources
arXiv2024

Exact a posteriori error control for variational problems via convex duality and explicit flux reconstruction

A posteriori error estimates are an important tool to bound discretization errors in terms of computable quantities avoiding regularity conditions that are often difficult to establish. For non-linear and non-differentiable problems, problems involving jumping coefficients, and finite element methods using anisotropic triangulations, such estimates often involve large factors, leading to sub-optimal error estimates. By making use of convex duality arguments, exact and explicit error representations are derived that avoid such effects.

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arXiv2024

Error analysis for a Crouzeix-Raviart approximation of the $p$-Dirichlet problem

In the present paper, we examine a Crouzeix-Raviart approximation for non-linear partial differential equations having a $(p,δ)$-structure for some $p\in (1,\infty)$ and $δ\ge 0$. We establish a priori error estimates, which are optimal for all $p\in (1,\infty)$ and $δ\ge 0$, medius error estimates, i.e., best-approximation results, and a primal-dual a posteriori error estimate, which is both reliable and efficient. The theoretical findings are supported by numerical experiments.

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arXiv2024

Error analysis for a Crouzeix-Raviart approximation of the variable exponent Dirichlet problem

In the present paper, we examine a Crouzeix-Raviart approximation of the $p(\cdot)$-Dirichlet problem. We derive a $\textit{medius}$ error estimate, $\textit{i.e.}$, a best-approximation result, which holds for uniformly continuous exponents and implies $\textit{a priori}$ error estimates, which apply for Hölder continuous exponents and are optimal for Lipschitz continuous exponents. Numerical experiments are carried out to review the theoretical findings.

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arXiv2024

Convergence analysis for a fully-discrete finite element approximation of the unsteady $p(\cdot,\cdot)$-Navier-Stokes equations

In the present paper, we establish the well-posedness, stability, and (weak) convergence of a fully-discrete approximation of the unsteady $p(\cdot,\cdot)$-Navier-Stokes equations employing an implicit Euler step in time and a discretely inf-sup-stable finite element approximation in space. Moreover, numerical experiments are carried out that supplement the theoretical findings.

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arXiv2023

Finite element discretization of the steady, generalized Navier-Stokes equations with inhomogeneous Dirichlet boundary conditions

We propose a finite element discretization for the steady, generalized Navier-Stokes equations for fluids with shear-dependent viscosity, completed with inhomogeneous Dirichlet boundary conditions and an inhomogeneous divergence constraint. We establish (weak) convergence of discrete solutions as well as a priori error estimates for the velocity vector field and the scalar kinematic pressure. Numerical experiments complement the theoretical findings.

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arXiv2023

On the stability and convergence of Discontinuous Galerkin schemes for incompressible flow

The property that the velocity $\boldsymbol{u}$ belongs to $L^\infty(0,T;L^2(Ω)^d)$ is an essential requirement in the definition of energy solutions of models for incompressible fluids. It is, therefore, highly desirable that the solutions produced by discretisation methods are uniformly stable in the $L^\infty(0,T;L^2(Ω)^d)$-norm. In this work, we establish that this is indeed the case for Discontinuous Galerkin (DG) discretisations (in time and space) of non-Newtonian models with $p$-structure, assuming that $p\geq \frac{3d+2}{d+2}$; the time discretisation is equivalent to the RadauIIA Implicit Runge-Kutta method. We also prove (weak) convergence of the numerical scheme to the weak solution of the system; this type of convergence result for schemes based on quadrature seems to be new. As an auxiliary result, we also derive Gagliardo-Nirenberg-type inequalities on DG spaces, which might be of independent interest.

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arXiv2023

Explicit a posteriori error representation for variational problems and application to TV-minimization

In this paper, we propose a general approach for explicit a posteriori error representation for convex minimization problems using basic convex duality relations. Exploiting discrete orthogonality relations in the space of element-wise constant vector fields as well as a discrete integration-by-parts formula between the Crouzeix-Raviart and the Raviart-Thomas element, all convex duality relations are transferred to a discrete level, making the explicit a posteriori error representation -- initially based on continuous arguments only -- practicable from a numerical point of view. In addition, we provide a generalized Marini formula for the primal solution that determines a discrete primal solution in terms of a given discrete dual solution. We benchmark all these concepts via the Rudin-Osher-Fatemi model. This leads to an adaptive algorithm that yields a (quasi-optimal) linear convergence rate.

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arXiv2023

A Local Discontinuous Galerkin approximation for the $p$-Navier-Stokes system, Part I: Convergence analysis

In the present paper, we propose a Local Discontinuous Galerkin (LDG) approximation for fully non-homogeneous systems of $p$-Navier-Stokes type. On the basis of the primal formulation, we prove well-posedness, stability (a priori estimates), and weak convergence of the method. To this end, we propose a new DG discretization of the convective term and develop an abstract non-conforming theory of pseudo-monotonicity, which is applied to our problem. We also use our approach to treat the $p$-Stokes problem.

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arXiv2022

Analysis of a fully-discrete, non-conforming approximation of evolution equations and applications

In this paper, we consider a fully-discrete approximation of an abstract evolution equation deploying a non-conforming spatial approximation and finite differences in time (Rothe-Galerkin method). The main result is the convergence of the discrete solutions to a weak solution of the continuous problem. Therefore, the result can be interpreted either as a justification of the numerical method or as an alternative way of constructing weak solutions. We formulate the problem in the very general and abstract setting of so-called non-conforming Bochner pseudo-monotone operators, which allows for a unified treatment of several evolution problems. Our abstract results for non-conforming Bochner pseudo-monotone operators allow to establish (weak) convergence just by verifying a few natural assumptions on the operators time-by-time and on the discretization spaces. Hence, applications and extensions to several other evolution problems can be performed easily. We exemplify the applicability of our approach on several DG schemes for the unsteady $p$-Navier-Stokes problem. The results of some numerical experiments are reported in the final section.

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arXiv2022

Convergence analysis of a Local Discontinuous Galerkin approximation for nonlinear systems with balanced Orlicz-structure

In this paper, we investigate a Local Discontinuous Galerkin (LDG) approximation for systems with balanced Orlicz-structure. We propose a new numerical flux, which yields optimal convergence rates for linear ansatz functions. In particular, our approach yields a unified treatment for problems with $(p,δ)$-structure for arbitrary $p \in (1,\infty)$ and $δ\ge 0$.

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arXiv2022

The Deep Ritz Method for Parametric $p$-Dirichlet Problems

We establish error estimates for the approximation of parametric $p$-Dirichlet problems deploying the Deep Ritz Method. Parametric dependencies include, e.g., varying geometries and exponents $p\in (1,\infty)$. Combining the derived error estimates with quantitative approximation theorems yields error decay rates and establishes that the Deep Ritz Method retains the favorable approximation capabilities of neural networks in the approximation of high dimensional functions which makes the method attractive for parametric problems. Finally, we present numerical examples to illustrate potential applications.

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arXiv2022

Explicit and efficient error estimation for convex minimization problems

We combine a systematic approach for deriving general a posteriori error estimates for convex minimization problems based on convex duality relations with a recently derived generalized Marini formula. The a posteriori error estimates are essentially constant-free and apply to a large class of variational problems including the $p$-Dirichlet problem, as well as degenerate minimization, obstacle and image de-noising problems. In addition, these a posteriori error estimates are based on a comparison to a given non-conforming finite element solution. For the $p$-Dirichlet problem, these a posteriori error bounds are equivalent to residual type a posteriori error bounds and, hence, reliable and efficient.

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arXiv2022

Error estimates for total-variation regularized minimization problems with singular dual solutions

Recent quasi-optimal error estimates for the finite element approximation of total-variation regularized minimization problems using the Crouzeix--Raviart finite element require the existence of a Lipschitz continuous dual solution, which is not generally given. We provide analytic proofs showing that the Lipschitz continuity of a dual solution is not necessary, in general. Using the Lipschitz truncation technique, we, in addition, derive error estimates that depend directly on the Sobolev regularity of a given dual solution.

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arXiv2021

Existence of steady solutions for a general model for micropolar electrorheological fluid flows

In this paper we study the existence of solutions to a steady system that describes the motion of a micropolar electrorheological fluid. The constitutive relations for the stress tensors belong to the class of generalized Newtonian fluids. The analysis of this particular problem leads naturally to weighted Sobolev spaces. By deploying the Lipschitz truncation technique, we establish the existence of solutions without additional assumptions on the electric field.

math.AP↗
arXiv2021

Existence of steady solutions for a model for micropolar electrorheological fluid flows with not globally $\log$--Hölder continuous shear exponent

In this paper, we study the existence of weak solutions to a steady system that describes the motion of a micropolar electrorheological fluid. The constitutive relations for the stress tensors belong to the class of generalized Newtonian fluids. The analysis of this particular problem leads naturally to weighted variable exponent Sobolev spaces. We establish the existence of solutions for a material function $\hat p$ that is $\log$--Hölder continuous and an electric field $\mathbf{E}$ for that $\vert\mathbf{E}\vert^2$ is bounded and smooth. Note that these conditions do not imply that the variable shear exponent $p=\hat p\circ\vert \mathbf{E}\vert^2$ is globally $\log$--Hölder continuous.

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