arXiv subjects

Norbert Heuer

Publications and source records attributed to Norbert Heuer.

At least 19 recordsLinked to original sources

arXiv2026

A DPG method for the circular arch problem

We consider an elastic model for a circular arch that incorporates membrane, transverse shear, and bending effects. The central line of the arch is partitioned into elements, and an ultra-weak variational formulation is developed alongside a discontinuous Petrov-Galerkin (DPG) approximation procedure based on so-called optimal test functions. The formulation uses discontinuous stress and displacement interpolations on the element mesh, with corresponding interface variables defined at the nodes. Theoretical analysis predicts optimal convergence rates for all quantities of interest, while also revealing potential error amplification influenced by the curvature of the arch and the imposed boundary conditions. The method is tested on examples with different support configurations. The numerical experiments confirm the theoretical predictions and further demonstrate that the accuracy of the DPG method can be improved by employing a suitably scaled test space norm.

math.NA↗

arXiv2026

Minimum residual discretization of a semilinear elliptic problem

We propose a least-squares penalization as a means to extend the discontinuous Petrov-Galerkin (DPG) method with optimal test functions to a class of semilinear elliptic problems. The nonlinear contributions are replaced with independent unknowns so that standard DPG techniques apply to the then linear problem with non-trivial kernel. The nonlinear relations are added as least-squares constraints. Assuming solvability of the semilinear problem and an Aubin-Nitsche-type approximation property for the primal variable, we prove a Cea estimate for the approximation error in canonical norms. Numerical results with uniform and adaptively refined meshes illustrate the performance of the scheme.

math.NA↗

arXiv2026

Boundary elements for clamped Kirchhoff--Love plates

We present a Galerkin boundary element method for clamped Kirchhoff--Love plates with piecewise smooth boundary. It is a direct method based on the representation formula and requires the inversion of the single-layer operator and an application of the double-layer operator to the Dirichlet data. We present trace approximation spaces of arbitrary order, required for both the Dirichlet data and the unknown Neumann trace. Our boundary element method is quasi-optimal with respect to the natural trace norm and achieves optimal convergence order under minimal regularity assumptions. We provide explicit representations of both boundary integral operators and discuss the implementation of the appearing integrals. Numerical experiments for smooth and non-smooth domains confirm predicted convergence rates.

math.NA↗

arXiv2025

Tensor finite elements for smectic liquid crystals

We present a tensor-based finite element scheme for a smectic-A liquid crystal model. We propose a simple C\'ea-type finite element projection in the linear case and prove its quasi-optimal convergence. Special emphasis is put on the formulation and treatment of appropriate boundary conditions. For the nonlinear case we present a formulation in two space dimensions and prove the existence of a solution. We propose a discretization that extends the linear case in Uzawa-fashion to the nonlinear case by an additional Poisson solver. Numerical results illustrate the performance and convergence of our schemes.

math.NA↗

arXiv2025

Normal-normal continuous symmetric stresses in mixed finite element elasticity

The classical continuous mixed formulation of linear elasticity with pointwise symmetric stresses allows for a conforming finite element discretization with piecewise polynomials of degree at least three. Symmetric stress approximations of lower polynomial order are only possible when their div-conformity is weakened to the continuity of normal-normal components. In two dimensions, this condition is meant pointwise along edges for piecewise polynomials, but a corresponding characterization for general piecewise H(div) tensors has been elusive. We introduce such a space and establish a continuous mixed formulation of linear planar elasticity with pointwise symmetric stresses that have, in a distributional sense, continuous normal-normal components across the edges of a shape-regular triangulation. The displacement is split into an $L_2$ field and a tangential trace on the skeleton of the mesh. The well-posedness of the new mixed formulation follows with a duality lemma relating the normal-normal continuous stresses with the tangential traces of displacements. For this new formulation we present a lowest-order conforming discretization. Stresses are approximated by piecewise quadratic symmetric tensors, whereas displacements are discretized by piecewise linear polynomials. The tangential displacement trace acts as a Lagrange multiplier and guarantees global div-conformity in the limit as the mesh-size tends to zero. We prove locking-free, quasi-optimal convergence of our scheme and illustrate this with numerical examples.

math.NA↗

arXiv2025

Normal-normal continuous symmetric stress approximation in three-dimensional linear elasticity

We present a conforming setting for a mixed formulation of linear elasticity with symmetric stress that has normal-normal continuous components across faces of tetrahedral meshes. We provide a stress element for this formulation with 30 degrees of freedom that correspond to standard boundary conditions. The resulting scheme converges quasi-optimally and is locking free. Numerical experiments illustrate the performance.

math.NA↗

arXiv2024

Generalized mixed and primal hybrid methods with applications to plate bending

We present an extended framework for hybrid finite element approximations of self-adjoint, positive definite operators. It covers the cases of primal, mixed, and ultraweak formulations, both at the continuous and discrete levels, and gives rise to conforming discretizations. Our framework allows for flexible continuity restrictions across elements, and includes the extreme cases of conforming and discontinuous hybrid methods. We illustrate an application of the framework to the Kirchhoff-Love plate pending model and present three primal hybrid and two mixed hybrid methods, four of them with numerical examples. In particular, we present conforming frameworks for (in classical meaning) non-conforming elements of Morley, Zienkiewicz triangular, and Hellan-Herrmann-Johnson types.

math.NA↗

arXiv2024

A fractional-order trace-dev-div inequality

The trace-dev-div inequality in $H^s$ controls the trace in the norm of $H^s$ by that of the deviatoric part plus the $H^{s-1}$ norm of the divergence of a quadratic tensor field different from the constant unit matrix. This is well known for $s=0$ and established for orders $0\le s\le 1$ and arbitrary space dimension in this note. For mixed and least-squares finite element error analysis in linear elasticity, this inequality allows to establish robustness with respect to the Lam\'e parameter $\lambda$.

math.NA↗

arXiv2023

A balanced finite-element method for an axisymmetrically loaded thin shell

We analyse a finite-element discretisation of a differential equation describing an axisymmetrically loaded thin shell. The problem is singularly perturbed when the thickness of the shell becomes small. We prove robust convergence of the method in a balanced norm that captures the layers present in the solution. Numerical results confirm our findings.

math.NA↗

arXiv2023

Mixed finite elements for Kirchhoff-Love plate bending

We present a mixed finite element method with triangular and parallelogram meshes for the Kirchhoff-Love plate bending model. Critical ingredient is the construction of low-dimensional local spaces and appropriate degrees of freedom that provide conformity in terms of a sufficiently large tensor space and that allow for any kind of physically relevant Dirichlet and Neumann boundary conditions. For Dirichlet boundary conditions and polygonal plates, we prove quasi-optimal convergence of the mixed scheme. An a posteriori error estimator is derived for the special case of the biharmonic problem. Numerical results for regular and singular examples illustrate our findings. They confirm expected convergence rates and exemplify the performance of an adaptive algorithm steered by our error estimator.

math.NA↗

arXiv2023

Robust DPG Fortin operators

At the fully discrete setting, stability of the discontinuous Petrov--Galerkin (DPG) method with optimal test functions requires local test spaces that ensure the existence of Fortin operators. We construct such operators for $H^1$ and $\boldsymbol{H}(\mathrm{div})$ on simplices in any space dimension and arbitrary polynomial degree. The resulting test spaces are smaller than previously analyzed cases. For parameter-dependent norms, we achieve uniform boundedness by the inclusion of exponential layers. As an example, we consider a canonical DPG setting for reaction-dominated diffusion. Our test spaces guarantee uniform stability and quasi-optimal convergence of the scheme. We present numerical experiments that illustrate the loss of stability and error control by the residual for small diffusion coefficient when using standard polynomial test spaces, whereas we observe uniform stability and error control with our construction.

math.NA↗

arXiv2023

A DPG method for the quad-curl problem

We derive an ultraweak variational formulation of the quad-curl problem in two and three dimensions. We present a discontinuous Petrov-Galerkin (DPG) method for its approximation and prove its quasi-optimal convergence. We illustrate how this method can be applied to the Stokes problem in two dimensions, after an application of the curl operator to eliminate the pressure variable. In this way, DPG techniques known from Kirchhoff-Love plates can be used. We present an a priori error estimate that improves a previous approximation result for effective shear forces by using a less restrictive regularity assumption. Numerical experiments illustrate our findings.

math.NA↗

arXiv2022

A DPG method for Reissner-Mindlin plates

We present a discontinuous Petrov-Galerkin (DPG) method with optimal test functions for the Reissner-Mindlin plate bending model. Our method is based on a variational formulation that utilizes a Helmholtz decomposition of the shear force. It produces approximations of the primitive variables and the bending moments. For any canonical selection of boundary conditions the method converges quasi-optimally. In the case of hard-clamped convex plates, we prove that the lowest-order scheme is locking free. Several numerical experiments confirm our results.

math.NA↗

arXiv2021

DPG methods for a fourth-order div problem

We study a fourth-order div problem and its approximation by the discontinuous Petrov-Galerkin method with optimal test functions. We present two variants, based on first and second-order systems. In both cases we prove well-posedness of the formulation and quasi-optimal convergence of the approximation. Our analysis includes the fully-discrete schemes with approximated test functions, for general dimension and polynomial degree in the first-order case, and for two dimensions and lowest-order approximation in the second-order case. Numerical results illustrate the performance for quasi-uniform and adaptively refined meshes.

math.NA↗

arXiv2021

MINRES for second-order PDEs with singular data

Minimum residual methods such as the least-squares finite element method (FEM) or the discontinuous Petrov--Galerkin method with optimal test functions (DPG) usually exclude singular data, e.g., non square-integrable loads. We consider a DPG method and a least-squares FEM for the Poisson problem. For both methods we analyze regularization approaches that allow the use of $H^{-1}$ loads, and also study the case of point loads. For all cases we prove appropriate convergence orders. We present various numerical experiments that confirm our theoretical results. Our approach extends to general well-posed second-order problems.

math.NA↗

arXiv2021

A DPG method for shallow shells

We develop and analyze a discontinuous Petrov--Galerkin method with optimal test functions (DPG method) for a shallow shell model of Koiter type. It is based on a uniformly stable ultraweak formulation and thus converges robustly quasi-uniformly. Numerical experiments for various cases, including the Scordelis--Lo cylindrical roof, elliptic and hyperbolic geometries, illustrate its performance. The built-in DPG error estimator gives rise to adaptive mesh refinements that are capable to resolve boundary and interior layers. The membrane locking is dealt with by raising the polynomial degree only of the tangential displacement trace variable.

math.NA↗

arXiv2021

Analysis of backward Euler primal DPG methods

We analyse backward Euler time stepping schemes for the primal DPG formulation of a class of parabolic problems. Optimal error estimates are shown in the natural norm and in the $L^2$ norm of the field variable. For the heat equation the solution of our primal DPG formulation equals the solution of a standard Galerkin scheme and, thus, optimal error bounds are found in the literature. In the presence of advection and reaction terms, however, the latter identity is not valid anymore and the analysis of optimal error bounds requires to resort to elliptic projection operators. It is essential that these operators be projections with respect to the spatial part of the PDE, as in standard Galerkin schemes, and not with respect to the full PDE at a time step, as done previously.

math.NA↗

arXiv2020

A robust DPG method for large domains

We observe a dramatic lack of robustness of the DPG method when solving problems on large domains and where stability is based on a Poincaré-type inequality. We show how robustness can be re-established by using appropriately scaled test norms. As model cases we study the Poisson problem and the Kirchhoff--Love plate bending model, and also include fully discrete variants where optimal test functions are approximated. Numerical experiments for both model problems, including an-isotropic domains and mixed boundary conditions, confirm our findings.

math.NA↗