arXiv subjects

Sören Bartels

Publications and source records attributed to Sören Bartels.

At least 19 recordsLinked to original sources

arXiv2026

Self-avoiding fluid deformable surfaces

We propose a numerical method for fluid deformable surfaces governed by surface Stokes flow and Helfrich bending energy under active growth, aiming to model shape evolution of the epithelium in developmental processes. To prevent self-intersections, which commonly arise under large deformations or low enclosed volume to area ratios, we incorporate the nonlocal tangent-point energy to penalize non-embedded configurations. The resulting formulation is discretized using higher order surface finite elements, with a parallelizable assembly strategy for the nonlocal terms. To tailor mesh quality to the geometric evolution, we propose a curvature-adaptive mesh redistribution strategy that improves mesh resolution in regions of high curvature. Numerical examples include the discocyte-to-stomatocyte transition and the inversion of a sphere within a spherical confinement. Both demonstrate the robustness of the method in capturing large deformations, self-avoidance, symmetry-breaking and growth-induced morphology changes.

physics.flu-dyn↗

arXiv2026

Variational problems with gradient constraints: $\textit{A priori}$ and $\textit{a posteriori}$ error identities

In this paper, on the basis of a (Fenchel) duality theory on the continuous level, we derive an $\textit{a posteriori}$ error identity for arbitrary conforming approximations of a primal formulation and a dual formulation of variational problems involving gradient constraints. In addition, on the basis of a (Fenchel) duality theory on the discrete level, we derive an $\textit{a priori}$ error identity that applies to the approximation of the primal formulation using the Crouzeix-Raviart element and to the approximation of the dual formulation using the Raviart-Thomas element, and leads to error decay rates that are optimal with respect to the regularity of a dual solution.

math.NA↗

arXiv2026

Unconstrained Scheme for Geometrically Constrained Gradient Flows

In this paper, we study the approximation of gradient flows of harmonic maps, which serve as model problems for applications in micromagnetics, liquid crystals, and nonlinear plate bending. Harmonic maps are vector fields that are critical points of the Dirichlet energy subject to the constraint that the vector field be unit length pointwise. Most existing time-stepping schemes for gradient flows deal with the constraint by linearizing the unit length constraint at every step, which involves solving for the solution increment in the tangent space of the constraint. These schemes lead to robust control over the violation of the constraint, but require solving degenerate saddle point systems at every step that may be difficult to precondition. In this paper, we propose a scheme that first computes the unconstrained increment and then projects this increment pointwise onto the tangent space. With an additional stabilization, this scheme is energy stable under mild step size restrictions and provides robust control of the unit length constraint violation. Our new scheme only requires the solution of decoupled symmetric positive definite systems at every step, which translates to a large increase in computational efficiency. We also propose a computable a posteriori criterion and a variable time-stepping procedure that guarantee the stability of the scheme. We conclude with computational examples demonstrating the efficacy of the scheme, and present a computational extension of the scheme to nonlinear plate bending.

math.NA↗

arXiv2026

A $\operatorname{prox}$-Based Semi-Smooth Newton Method for Convex Variational Problems

In this paper, we devise a $\operatorname{prox}$-based semi-smooth Newton method that is applicable to a finite element discretization of a broad class of nonsmooth convex variational problems, including the TV-minimization problem, the $p$-Dirichlet problem, the obstacle problem, and the elasto-plastic torsion problem. To this end, on the basis of the proximity operator, the discrete primal-dual optimality conditions are reformulated as nonlinear operator equations with Newton-differentiable structure. Under suitable assumptions on the energy densities, we establish the global well-posedness and local super-linear convergence of the resulting semi-smooth Newton method. The proposed approach coincides with established semi-smooth Newton methods for obstacle-type problems, satisfies a primal-dual invariance, and, under suitable additional assumptions, is globally well-posed in the infinite-dimensional setting.

math.OC↗

arXiv2026

A $\operatorname{prox}$-Based Semi-Smooth Newton Method for TV-Minimization

In this paper, we devise a $\operatorname{prox}$-based semi-smooth Newton method for the non-differentiable TV-minimization problem. To this end, the primal-dual optimality conditions are reformulated as a nonlinear operator equation with Newton-(type-)differentiable structure. We investigate the question of well-posedness of the resulting semi-smooth Newton scheme in the infinite-dimensional setting and identify structural properties of the associated Newton-type derivatives. For a conforming finite element discretization, we prove that the resulting semi-smooth Newton method is globally well-posed and locally super-linearly convergent. The approach extends to a large class of convex minimization problems, coincides with established semi-smooth Newton methods for obstacle problems, satisfies a primal-dual invariance, and, under suitable additional assumptions, is well-posed in the infinite-dimensional setting. Numerical experiments indicate a robust practical performance of the proposed method, including reliable reduction of the discrete primal-dual gap estimator to machine precision, robustness with respect to the choice of proximity parameters, an improved convergence basin compared to a canonical primal semi-smooth Newton method, and effective performance even for quadratically graded meshes using only a mesh-independent initialization criterion.

math.NA↗

arXiv2026

Projection-free approximation of flows of harmonic maps with quadratic constraint accuracy and variable step sizes

We construct and analyze a projection-free linearly implicit method for the approximation of flows of harmonic maps into spheres. The proposed method is unconditionally energy stable and, under a sharp discrete regularity condition, achieves second-order accuracy with respect to the constraint violation. Furthermore, the method accommodates variable step sizes to speed up the convergence to stationary points and to improve the accuracy of the numerical solutions near singularities, without affecting the unconditional energy stability and the constraint violation property. We illustrate the accuracy in approximating the unit-length constraint and the performance of the method through a series of numerical experiments, and compare it with the linearly implicit Euler and two-step BDF methods.

math.NA↗

arXiv2025

Babuška's paradox in a nonlinear bending-folding model

The Babuška or plate paradox concerns the failure of convergence when a domain with curved boundary is approximated by polygonal domains in linear bending problems with simply supported boundary conditions. It can be explained via a boundary integral representation of the total Gaussian curvature that is part of the Kirchhoff--Love bending energy. It is shown that the paradox also occurs for a nonlinear bending-folding model which enforces vanishing Gaussian curvature. A simple remedy that is compatible with simplicial finite element methods to avoid incorrect convergence is devised.

math.NA↗

arXiv2025

Quasi-optimal error estimates for the approximation of stable stationary states of the elastic energy of inextensible curves

We establish local existence and a quasi-optimal error estimate for piecewise cubic minimizers to the bending energy under a discretized inextensibility constraint. In previous research a discretization is used where the inextensibility constraint is only enforced at the nodes of the discretization. We show why this discretization leads to suboptimal convergence rates and we improve on it by also enforcing the constraint in the midpoints of each subinterval. We then use the inverse function theorem to prove existence and an error estimate for stationary states of the bending energy that yields quasi-optimal convergence. We use numerical simulations to verify the theoretical results experimentally.

math.NA↗

arXiv2025

Quasi-optimal error estimate for the approximation of the elastic flow of inextensible curves

A space-discretization for the elastic flow of inextensible curves is devised and quasi-optimal convergence of the corresponding semi-discrete problem is proved for a suitable discretization of the nonlinear inextensibility constraint. Further a fully discrete time-stepping scheme that incorporates this constraint is proposed and unconditional stability and convergence of the discrete scheme are proved. Finally some numerical simulations are used to verify the obtained results experimentally.

math.NA↗

arXiv2025

Error analysis for a Crouzeix-Raviart approximation of the obstacle problem

In the present paper, we study a Crouzeix-Raviart approximation of the obstacle problem, which imposes the obstacle constraint in the midpoints (i.e., barycenters) of the elements of a triangulation. We establish a priori error estimates imposing natural regularity assumptions, which are optimal, and the reliability and efficiency of a primal-dual type a posteriori error estimator for general obstacles and involving data oscillation terms stemming only from the right-hand side. Numerical experiments are carried out to support the theoretical findings.

math.NA↗

arXiv2025

Numerical simulation of a fine-tunable Föppl-von Kármán model for foldable and bilayer plates

A numerical scheme is proposed to identify low energy configurations of a Föppl-von Kármán model for bilayer plates. The dependency of the corresponding elastic energy on the in-plane displacement $u$ and the out-of-plane deflection $w$ leads to a practical minimization of the functional via a decoupled gradient flow. In particular, the energies of the resulting iterates are shown to be monotonically decreasing. The discretization of the model relies on $P1$ finite elements for the horizontal part $u$ and utilizes the discrete Kirchhoff triangle for the vertical component $w$. The model allows for analysing various different problem settings via numerical simulation: (i) stable low-energy configurations are detected dependent on a specified prestrain described by elastic material properties, (ii) curvature inversions of spherical and cylindrical configurations are investigated, (iii) elastic responses of foldable cardboards for different spontaneous curvatures and crease geometries are compared.

math.NA↗

arXiv2024

Benchmarking Numerical Algorithms for Harmonic Maps into the Sphere

We numerically benchmark methods for computing harmonic maps into the unit sphere, with particular focus on harmonic maps with singularities. For the discretization we compare two different approaches, both based on Lagrange finite elements. While the first method enforces the unit-length constraint only at the Lagrange nodes, the other one adds a pointwise projection to fulfill the constraint everywhere. For the solution of the resulting algebraic problems we compare a nonconforming gradient flow with a Riemannian trust-region method. Both are energy-decreasing and can be shown to converge globally to stationary points of the discretized Dirichlet energy. We observe that while the nonconforming and the conforming discretizations both show similar behavior for smooth problems, the nonconforming discretization handles singularities better. On the solver side, the second-order trust-region method converges after few steps, whereas the number of gradient-flow steps increases proportionally to the inverse grid element diameter.

math.NA↗

arXiv2024

Optimization of an eigenvalue arising in optimal insulation with a lower bound

An eigenvalue problem arising in optimal insulation related to the minimization of the heat decay rate of an insulated body is adapted to enforce a positive lower bound imposed on the distribution of insulating material. We prove the existence of optimal domains among a class of convex shapes and propose a numerical scheme to approximate the eigenvalue. The stability of the shape optimization among convex, bounded domains in $\mathbb{R}^3$ is proven for an approximation with polyhedral domains under a non-conformal convexity constraint. We prove that on the ball, symmetry breaking of the optimal insulation can be expected in general. To observe how the lower bound affects the breaking of symmetry in the optimal insulation and the shape optimization, the eigenvalue and optimal domains are approximated for several values of mass $m$ and lower bounds $\ell_{\min}\ge0$. The numerical experiments suggest, that in general symmetry breaking still arises, unless $m$ is close to a critical value $m_0$, and $\ell_{\min}$ large enough such that almost all of the mass $m$ is fixed through the lower bound. For $\ell_{\min}=0$, the numerical results are consistent with previous numerical experiments on shape optimization restricted to rotationally symmetric, convex domains.

math.NA↗

arXiv2024

$\textit{A priori}$ and $\textit{a posteriori}$ error identities for the scalar Signorini problem

In this paper, on the basis of a (Fenchel) duality theory on the continuous level, we derive an $\textit{a posteriori}$ error identity for arbitrary conforming approximations of the primal formulation and the dual formulation of the scalar Signorini problem. In addition, on the basis of a (Fenchel) duality theory on the discrete level, we derive an $\textit{a priori}$ error identity that applies to the approximation of the primal formulation using the Crouzeix-Raviart element and to the approximation of the dual formulation using the Raviart-Thomas element, and leads to quasi-optimal error decay rates without imposing additional assumptions on the contact set and in arbitrary space dimensions.

math.NA↗

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.

math.NA↗

arXiv2024

Quadratic constraint consistency in the projection-free approximation of harmonic maps and bending isometries

We devise a projection-free iterative scheme for the approximation of harmonic maps that provides a second-order accuracy of the constraint violation and is unconditionally energy stable. A corresponding error estimate is valid under a mild but necessary discrete regularity condition. The method is based on the application of a BDF2 scheme and the considered problem serves as a model for partial differential equations with holonomic constraint. The performance of the method is illustrated via the computation of stationary harmonic maps and bending isometries.

math.NA↗

arXiv2024

Necessary and Sufficient Conditions for Avoiding Babuska's Paradox on Simplicial Meshes

It is shown that discretizations based on variational or weak formulations of the plate bending problem with simple support boundary conditions do not lead to failure of convergence when polygonal domain approximations are used and the imposed boundary conditions are compatible with the nodal interpolation of the restriction of certain regular functions to approximating domains. It is further shown that this is optimal in the sense that a full realization of the boundary conditions leads to failure of convergence for conforming methods. The abstract conditions imply that standard nonconforming and discontinuous Galerkin methods converge correctly while conforming methods require a suitable relaxation of the boundary condition. The results are confirmed by numerical experiments.

math.NA↗

arXiv2023

Numerical Approximation of Optimal Convex Shapes in $\mathbb{R}^3$

In the optimization of convex domains under a PDE constraint numerical difficulties arise in the approximation of convex domains in $\mathbb{R}^3$. Previous research used a restriction to rotationally symmetric domains to reduce shape optimization problems to a two-dimensional setting. In the current research, two approaches for the approximation in $\mathbb{R}^3$ are considered. First, a notion of discrete convexity allows for a nearly convex approximation with polyhedral domains. An alternative approach is based on the recent observation that higher order finite elements can approximate convex functions conformally. As a second approach these results are used to approximate optimal convex domains with isoparametric convex domains. The proposed algorithms were tested on shape optimization problems constrained by a Poisson equation and both algorithms achieved similar results.

math.NA↗