arXiv subjects

Joachim Schöberl

Publications and source records attributed to Joachim Schöberl.

At least 19 recordsLinked to original sources

arXiv2026

High order biorthogonal functions for the discrete de Rham complex on simplices*

It is well known that the choice of basis functions in hp-FEM heavily influences the stability and the computational cost in order to obtain an approximate solution. For simplicial elements in two and three space dimensions, tensor-product-like basis functions built from Jacobi polynomials with different weights yield optimal properties with respect to condition number and sparsity. In this paper we construct such a high order basis for the Nédélec spaces and for the Raviart-Thomas and Brezzi-Douglas-Marini spaces, modifying existing definitions of Zaglmayr so that the functions belong to the Nédélec spaces of first kind and the Raviart-Thomas spaces. The bases are designed so that the Nédélec space of second kind and the BDM space are extensions of the respective other spaces. In the second part of the paper we introduce biorthogonal basis functions for $H(\text{div}, Ω)$ continuing previous research for $H^1(Ω)$ and $H(\text{curl}, Ω)$. These functions can be expressed in closed form as sums of tensor products of Jacobi polynomials, which allows for fast computation of the $L^2$-projection.

math.NA↗

arXiv2026

A higher-order dual cell method for time-domain Maxwell equations

We present a higher-order extension of the dual cell method for the time-domain Maxwell equations in three spatial dimensions. The approach builds upon a variational reinterpretation of the Finite Integration Technique on dual meshes and generalises a previously developed two-dimensional high-order formulation. The electric and magnetic fields are discretised on mutually dual barycentric grids using curl-conforming polynomial spaces constructed via tensor-product Gauss--Radau interpolation. The resulting semi-discrete formulation yields block-diagonal mass matrices and sparse discrete curl operators, enabling explicit time integration while preserving a discrete energy identity. Special attention is devoted to the construction of compatible approximation spaces on the three-dimensional primal and dual meshes, the reference-to-physical element mappings, and the preservation of tangential continuity. We show that the method achieves arbitrary-order convergence, avoids spurious modes, and maintains optimal sparsity properties. Numerical experiments confirm spectral correctness, high-order accuracy, and computational efficiency on unstructured tetrahedral meshes.

math.NA↗

arXiv2026

Generalizing Riemann curvature to Regge metrics

In this paper, we propose a generalization of the Riemann curvature tensor on manifolds (of dimension two or higher) endowed with a Regge metric. Specifically, while all components of the metric tensor are assumed to be smooth within elements of a triangulation of the manifold, they need not be smooth across element interfaces, where only continuity of the tangential components is assumed. While linear derivatives of the metric can be generalized to Schwartz distributions, similarly generalizing the classical Riemann curvature tensor, a nonlinear second-order derivative of the metric, requires more care. We propose a generalization that combines the classical angle defect and jumps in the second fundamental form across element interfaces, and argue its correctness. Specifically, if a piecewise smooth metric approximates a globally smooth metric, then our generalized Riemann curvature tensor approximates the classical Riemann curvature tensor associated with the latter. Moreover, we show that if the metric approximation converges at some rate in a mesh-dependent norm equivalent to the $L^2$ norm, then the curvature approximation converges in the negative Sobolev space $H^{-2}$, the dual space of $H^2_0$, at the same rate, under additional assumptions. By appropriate contractions of the generalized Riemann curvature tensor, this work also provides generalizations of scalar curvature, the Ricci curvature tensor, and the Einstein tensor in any dimension.

math.NA↗

arXiv2025

Two-level additive Schwarz preconditioners for reduced integration methods

Incompressible fluid flow problems appear frequently in different applications. The discretization of such problems may result in large and ill-conditioned systems of linear equations. We consider the case of the Stokes equations discretized using a reduced integration method which approximates the incompressibility constraint by a penalty term thus allowing the problem to be solved only in terms of the velocity unknowns. We investigate the numerical scalability of a two-level overlapping additive Schwarz method with a reduced dimension generalized Dryja-Smith-Widlund (RGDSW) coarse space. In addition, we discuss the parallel implementation of the examples using the Fast and Robust Overlapping Schwarz (FROSch) package for additive Schwarz preconditioners and the NGSolve library, which implements multiple finite element space formulations.

math.NA↗

arXiv2025

A four-field mixed formulation for incompressible finite elasticity

In this work, we generalize the mass-conserving mixed stress (MCS) finite element method for Stokes equations [Gopalakrishnan J., Lederer P., and Schöberl J., A mass conserving mixed stress formulation for the Stokes equations, IMA Journal of Numerical Analysis 40(3), 1838-1874 (2019)], involving normal velocity and tangential-normal stress continuous fields, to incompressible finite elasticity. By means of the three-field Hu-Washizu principle, introducing the displacement gradient and 1st Piola-Kirchhoff stress tensor as additional fields, we circumvent the inversion of the constitutive law. We lift the arising distributional derivatives of the displacement gradient to a regular auxiliary displacement gradient field. Static condensation can be applied at the element level, providing a global pure displacement problem to be solved. We present a stabilization motivated by Hybrid Discontinuous Galerkin methods. A solving algorithm is discussed, which asserts the solvability of the arising linearized subproblems for problems with physically positive eigenvalues. The excellent performance of the proposed method is corroborated by several numerical experiments.

math.NA↗

arXiv2025

A 2-complex containing Sobolev spaces of matrix fields

Using a generalization of complexes, called 2-complexes, this paper defines and analyzes new Sobolev spaces of matrix fields and their interrelationships within a commuting diagram. These spaces have very weak second-order derivatives. An example is the space of matrix fields of square-integrable components whose row-wise divergence followed by yet another divergence operation yield a function in a standard negative-order Sobolev space. Similar spaces where the double divergence is replaced by a curl composed with divergence, or a double curl operator (the incompatibility operator), are also studied. Stable decompositions of such spaces in terms of more regular component functions (which are continuous in natural norms) are established. Appropriately ordering such Sobolev spaces with and without boundary conditions (in a weak sense), we discover duality relationships between them. Motivation to study such Sobolev spaces, from a finite element perspective and implications for weak well-posed variational formulations are pointed out.

math.AP↗

arXiv2024

High-order projection-based upwind method for simulation of transitional turbulent flows

We present a scalable, high-order implicit large-eddy simulation (ILES) approach for incompressible transitional flows. This method employs the mass-conserving mixed stress (MCS) method for discretizing the Navier-Stokes equations. The MCS method's low dissipation characteristics, combined with the introduced operator-splitting solution technique, result in a high-order solver optimized for efficient and parallel computation of under-resolved turbulent flows. We further enhance the inherent capabilities of the ILES model by incorporating high-order upwind fluxes and are examining its approximation behaviour in transitional aerodynamic flow problems. In this study, we use flows over the Eppler 387 airfoil at Reynolds numbers up to $3 \cdot 10^5$ as benchmarks for our simulations.

cs.CE↗

arXiv2024

On the improved convergence of lifted distributional Gauss curvature from Regge elements

Although Regge finite element functions are not continuous, useful generalizations of nonlinear derivatives like the curvature, can be defined using them. This paper is devoted to studying the convergence of the finite element lifting of a generalized (distributional) Gauss curvature defined using a metric tensor approximation in the Regge finite element space. Specifically, we investigate the interplay between the polynomial degree of the curvature lifting by Lagrange elements and the degree of the metric tensor in the Regge finite element space. Previously, a superconvergence result, where convergence rate of one order higher than expected, was obtained when the approximate metric is the canonical Regge interpolant of the exact metric. In this work, we show that an even higher order can be obtained if the degree of the curvature lifting is reduced by one polynomial degree and if at least linear Regge elements are used. These improved convergence rates are confirmed by numerical examples.

math.NA↗

arXiv2024

The Hellan-Herrmann-Johnson and TDNNS method for linear and nonlinear shells

In this paper we extend the recently introduced mixed Hellan-Herrmann-Johnson (HHJ) method for nonlinear Koiter shells to nonlinear Naghdi shells by means of a hierarchical approach. The additional shearing degrees of freedom are discretized by H(curl)-conforming Nédélec finite elements entailing a shear locking free method. By linearizing the models we obtain in the small strain regime linear Kirchhoff-Love and Reissner-Mindlin shell formulations, which reduce for plates to the originally proposed HHJ and TDNNS method for Kirchhoff-Love and Reissner-Mindlin plates, respectively. By using the Regge interpolation operator we obtain locking-free arbitrary order shell methods. Additionally, the methods can be directly applied to structures with kinks and branched shells. Several numerical examples and experiments are performed validating the excellence performance of the proposed shell elements.

math.NA↗

arXiv2023

Mass lumping the dual cell method to arbitrary polynomial degree for acoustic and electromagnetic waves

We present a fundamental improvement of a high polynomial degree time domain cell method recently introduced by the last three authors. The published work introduced a method featuring block-diagonal system matrices where the block size and conditioning scaled poorly with respect to polynomial degree. The issue is herein bypassed by the construction of new basis functions exploiting quadrature rule based mass lumping techniques for arbitrary polynomial degrees in two dimensions for the Maxwell equations and the acoustic wave equation in the first order velocity pressure formulation. We characterize the degrees of freedom of all new discrete approximation spaces we employ for differential forms and show that the resulting block diagonal (inverse) mass matrices have block sizes independent of the polynomial degree. We demonstrate on an extensive number of examples how the new technique is applicable and efficient for large scale computations.

math.NA↗

arXiv2023

A discontinuous Galerkin approach for atmospheric flows with implicit condensation

We present a discontinuous Galerkin method for moist atmospheric dynamics, with and without warm rain. By considering a combined density for water vapour and cloud water, we avoid the need to model and compute a source term for condensation. We recover the vapour and cloud densities by solving a pointwise non-linear problem each time step. Consequently, we enforce the requirement for the water vapour not to be supersaturated implicitly. Together with an explicit time-stepping scheme, the method is highly parallelisable and can utilise high-performance computing hardware. Furthermore, the discretisation works on structured and unstructured meshes in two and three spatial dimensions. We illustrate the performance of our approach using several test cases in two and three spatial dimensions. In the case of a smooth, exact solution, we illustrate the optimal higher-order convergence rates of the method.

math.NA↗

arXiv2023

High order biorthogonal functions in H(Curl)

From the literature, it is known that the choice of basis functions in hp-FEM heavily influences the computational cost in order to obtain an approximate solution. Depending on the choice of the reference element, suitable tensor product like basis functions of Jacobi polynomials with different weights lead to optimal properties due to condition number and sparsity. This paper presents biorthogonal basis functions to the primal basis functions mentioned above. The authors investigate hypercubes and simplices as reference elements, as well as the cases of $H^1$ and H(Curl). The functions can be expressed sums of tensor products of Jacobi polynomials with maximal two summands.

math.NA↗

arXiv2023

Error Analysis of an HDG Method with Impedance Traces for the Helmholtz Equation

In this work, a novel analysis of a hybrid discontinuous Galerkin method for the Helmholtz equation is presented. It uses wavenumber, mesh size and polynomial degree independent stabilisation parameters leading to impedance traces between elements. With analysis techniques based on projection operators unique discrete solvability without a resolution condition and optimal convergence rates with respect to the mesh size are proven. The considered method is tailored towards enabling static condensation and the usage of iterative solvers.

math.NA↗

arXiv2023

Divergence-conforming velocity and vorticity approximations for incompressible fluids obtained with minimal facet coupling

We introduce two new lowest order methods, a mixed method, and a hybrid Discontinuous Galerkin (HDG) method, for the approximation of incompressible flows. Both methods use divergence-conforming linear Brezzi-Douglas-Marini space for approximating the velocity and the lowest order Raviart-Thomas space for approximating the vorticity. Our methods are based on the physically correct viscous stress tensor of the fluid, involving the symmetric gradient of velocity (rather than the gradient), provide exactly divergence-free discrete velocity solutions, and optimal error estimates that are also pressure robust. We explain how the methods are constructed using the minimal number of coupling degrees of freedom per facet. The stability analysis of both methods are based on a Korn-like inequality for vector finite elements with continuous normal component. Numerical examples illustrate the theoretical findings and offer comparisons of condition numbers between the two new methods.

math.NA↗

arXiv2023

High-order projection-based upwind method for implicit large eddy simulation

We assess the ability of three different approaches based on high-order discontinuous Galerkin methods to simulate under-resolved turbulent flows. The capabilities of the mass conserving mixed stress method as structure resolving large eddy simulation solver are examined. A comparison of a variational multiscale model to no-model or an implicit model approach is presented via numerical results. In addition, we present a novel approach for turbulent modeling in wall-bounded flows. This new technique provides a more accurate representation of the actual subgrid scales in the near wall region and gives promising results for highly under-resolved flow problems. In this paper, the turbulent channel flow and periodic hill flow problem are considered as benchmarks for our simulations.

physics.flu-dyn↗

arXiv2022

Analysis of curvature approximations via covariant curl and incompatibility for Regge metrics

The metric tensor of a Riemannian manifold can be approximated using Regge finite elements and such approximations can be used to compute approximations to the Gauss curvature and the Levi-Civita connection of the manifold. It is shown that certain Regge approximations yield curvature and connection approximations that converge at a higher rate than previously known. The analysis is based on covariant (distributional) curl and incompatibility operators which can be applied to piecewise smooth matrix fields whose tangential-tangential component is continuous across element interfaces. Using the properties of the canonical interpolant of the Regge space, we obtain superconvergence of approximations of these covariant operators. Numerical experiments further illustrate the results from the error analysis.

math.NA↗

arXiv2022

Curing spurious magneto-mechanical coupling in soft non-magnetic materials

The present work is concerned with the issue of spurious coupling effects that are pervasive in fully coupled magneto-mechanical finite element simulations involving very soft non-magnetic or air-like media. We first address the characterization of the spurious magneto-mechanical effects and their intuitive interpretation based on energy considerations. Then, as main contribution, we propose two new ways to prune the undesired spurious magneto-mechanical coupling in non-magnetic media. The proposed methods are compared with established methods in the context of magnetic bodies embedded in (i) air or vacuum and (ii) very soft elastic non-magnetic media. The comparison shows that the proposed approaches are accurate and effective. They, furthermore, allow for a consistent linearization of the coupled boundary value problems, which is crucial for the simulation of compliant structures. For reproducibility and accessibility of the proposed methods, we provide our implementations with Netgen/NGSolve as well as all codes necessary for the reproduction of our results as supplementary material.

math.NA↗

arXiv2022

Numerical shape optimization of the Canham-Helfrich-Evans bending energy

In this paper we propose a novel numerical scheme for the Canham-Helfrich-Evans bending energy based on a three-field lifting procedure of the distributional shape operator to an auxiliary mean curvature field. Together with its energetic conjugate scalar stress field as Lagrange multiplier the resulting fourth order problem is circumvented and reduced to a mixed saddle point problem involving only second order differential operators. Further, we derive its analytical first variation (also called first shape derivative), which is valid for arbitrary polynomial order, and discuss how the arising shape derivatives can be computed automatically in the finite element software NGSolve. We finish the paper with several numerical simulations showing the pertinence of the proposed scheme and method.

math.NA↗