arXiv subjects

Thomas Führer

Publications and source records attributed to Thomas Führer.

At least 19 recordsLinked to original sources

arXiv2026

Space-time finite element interpolation of nonsmooth functions satisfying boundary conditions

We construct Scott-Zhang-type space-time quasi-interpolation operators on tensor- product meshes for parabolic settings. Specifically, these are linear and uniformly bounded projections from the classical parabolic energy space onto a space-time tensor-product finite element space that preserve discrete Dirichlet boundary conditions on the lateral space-time boundary. We apply our operator in the context of space-time first-order system least-squares finite elements for parabolic equations to establish a quasi-optimal method for inhomogeneous Dirichlet boundary conditions.

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arXiv2026

Adaptive finite element methods with optimally preconditioned GMRES guarantee optimal complexity

We analyze optimal complexity of adaptive finite element methods (AFEMs) for general second-order linear elliptic partial differential equations (PDEs) in the Lax-Milgram setting. To this end, we formulate an adaptive algorithm which steers the local mesh-refinement as well as the termination of a generalized minimal residual solver (GMRES) with optimal preconditioner to solve the arising non-symmetric finite element systems. Algorithmic interplay of mesh-refinement and iterative solver is shown to be optimal: A natural and fully computable quasi-error monitoring discretization error and algebraic solver error guarantees unconditional convergence for any choice of adaptivity parameters, i.e., the algorithm cannot fail to converge. This is ensured algorithmically via a novel adaptive feedback-control for the solver-termination parameter that monitors and ensures full R-linear convergence. Finally, the quasi-error even decays with optimal rates with respect to the overall computational complexity if the adaptivity parameters are chosen sufficiently small.

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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.

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arXiv2025

Space-time FEM-BEM couplings for parabolic transmission problems

We develop couplings of a recent space-time first-order system least-squares (FOSLS) method for parabolic problems and space-time boundary element methods (BEM) for the heat equation to numerically solve a parabolic transmission problem on the full space and a finite time interval. In particular, we demonstrate coercivity of the couplings under certain restrictions and validate our theoretical findings by numerical experiments.

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arXiv2025

A posteriori analysis of neural network approximations

In a general setting, we study a posteriori estimates used in finite element analysis to measure the error between a solution and its approximation. The latter is not necessarily generated by a finite element method. We show that the error is equivalent to the sum of two residuals provided that the underlying variational formulation is well posed. The first contribution is the projection of the residual to a finite-dimensional space and is therefore computable, while the second one can be reliably estimated by a computable upper bound in many practical scenarios. Assuming sufficiently accurate quadrature, our findings can be used to estimate the error of, e.g., neural network outputs. Two important applications can be considered during optimization: first, the estimators are used to monitor the error in each solver step, or, second, the two estimators are included in the loss functional, and therefore provide control over the error. As a model problem, we consider a second-order elliptic partial differential equation and discuss different variational formulations thereof, including several options to include boundary conditions in the estimators. Various numerical experiments are presented to validate our findings.

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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 Céa-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.

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arXiv2025

A space-time finite element method for parabolic obstacle problems

We propose and analyze a general framework for space-time finite element methods that is based on least-squares finite element methods for solving a first-order reformulation of the thick parabolic obstacle problem. Discretizations based on simplicial and prismatic meshes are studied and we show a priori error estimates for both. Convergence rates are derived for sufficiently smooth solutions. Reliable a posteriori bounds are provided and used to steer an adaptive algorithm. Numerical experiments including a one-phase Stefan problem and an American option pricing problem are presented.

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arXiv2024

Aubin--Nitsche-type estimates for space-time FOSLS for parabolic PDEs

We develop Aubin--Nitsche-type estimates for recently proposed first-order system least-squares finite element methods (FOSLS) for the heat equation. Under certain assumptions, which are satisfied if the spatial domain is convex and the heat source and initial datum are sufficiently smooth, we prove that the $L^2$ error of approximations of the scalar field variable converges at a higher rate than the overall error. Furthermore, a higher-order conservation property is shown. In addition, we discuss quasi-optimality in weaker norms. Numerical experiments confirm our theoretical findings.

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arXiv2024

Mixed finite element methods for elliptic obstacle problems

Mixed variational formulations for the first-order system of the elastic membrane obstacle problem and the second-order system of the Kirchhoff--Love plate obstacle problem are proposed. The force exerted by the rigid obstacle is included as a new unknown. A priori and a posteriori error estimates are derived for both obstacle problems. The a posteriori error estimates are based on conforming postprocessed solutions. Numerical experiments conclude this work.

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arXiv2024

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.

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arXiv2024

Robust hybrid finite element methods for reaction-dominated diffusion problems

For a reaction-dominated diffusion problem we study a primal and a dual hybrid finite element method where weak continuity conditions are enforced by Lagrange multipliers. Uniform robustness of the discrete methods is achieved by enriching the local discretization spaces with modified face bubble functions which decay exponentially in the interior of an element depending on the ratio of the singular perturbation parameter and the local mesh-size. A posteriori error estimators are derived using Fortin operators. They are robust with respect to the singular perturbation parameter. Numerical experiments are presented that show that oscillations, if present, are significantly smaller then those observed in common finite element methods.

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arXiv2024

Quasi-interpolators with application to postprocessing in finite element methods

We design quasi-interpolation operators based on piecewise polynomial weight functions of degree less than or equal to $p$ that map into the space of continuous piecewise polynomials of degree less than or equal to $p+1$. We show that the operators have optimal approximation properties, i.e., of order $p+2$. This can be exploited to enhance the accuracy of finite element approximations provided that they are sufficiently close to the orthogonal projection of the exact solution on the space of piecewise polynomials of degree less than or equal to $p$. Such a condition is met by various numerical schemes, e.g., mixed finite element methods and discontinuous Petrov--Galerkin methods. Contrary to well-established postprocessing techniques which also require this or a similar closeness property, our proposed method delivers a conforming postprocessed solution that does not rely on discrete approximations of derivatives nor local versions of the underlying PDE. In addition, we introduce a second family of quasi-interpolation operators that are based on piecewise constant weight functions, which can be used, e.g., to postprocess solutions of hybridizable discontinuous Galerkin methods. Another application of our proposed operators is the definition of projection operators bounded in Sobolev spaces with negative indices. Numerical examples demonstrate the effectiveness of our approach.

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arXiv2023

Well-posedness of first-order acoustic wave equations and space-time finite element approximation

We study a first-order system formulation of the (acoustic) wave equation and prove that the operator of this system is an isomorphsim from an appropriately defined graph space to L^2. The results rely on well-posedness and stability of the weak and ultraweak formulation of the second-order wave equation. As an application we define and analyze a space-time least-squares finite element method for solving the wave equation. Some numerical examples for one- and two- dimensional spatial domains are presented.

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arXiv2023

A DPG method for linear quadratic optimal control problems

The DPG method with optimal test functions for solving linear quadratic optimal control problems with control constraints is studied. We prove existence of a unique optimal solution of the nonlinear discrete problem and characterize it through first order optimality conditions. Furthermore, we systematically develop a priori as well as a posteriori error estimates. Our proposed method can be applied to a wide range of constrained optimal control problems subject to, e.g., scalar second-order PDEs and the Stokes equations. Numerical experiments that illustrate our theoretical findings are presented.

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arXiv2023

Least-squares finite elements for distributed optimal control problems

We provide a framework for the numerical approximation of distributed optimal control problems, based on least-squares finite element methods. Our proposed method simultaneously solves the state and adjoint equations and is $\inf$--$\sup$ stable for any choice of conforming discretization spaces. A reliable and efficient a posteriori error estimator is derived for problems where box constraints are imposed on the control. It can be localized and therefore used to steer an adaptive algorithm. For unconstrained optimal control problems, i.e., the set of controls being a Hilbert space, we obtain a coercive least-squares method and, in particular, quasi-optimality for any choice of discrete approximation space. For constrained problems we derive and analyze a variational inequality where the PDE part is tackled by least-squares finite element methods. We show that the abstract framework can be applied to a wide range of problems, including scalar second-order PDEs, the Stokes problem, and parabolic problems on space-time domains. Numerical examples for some selected problems are presented.

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arXiv2023

On a mixed FEM and a FOSLS with $H^{-1}$ loads

We study variants of the mixed finite element method (mixed FEM) and the first-order system least-squares finite element (FOSLS) for the Poisson problem where we replace the load by a suitable regularization which permits to use $H^{-1}$ loads. We prove that any bounded $H^{-1}$ projector onto piecewise constants can be used to define the regularization and yields quasi-optimality of the lowest-order mixed FEM resp. FOSLS in weaker norms. Examples for the construction of such projectors are given. One is based on the adjoint of a weighted Clément quasi-interpolator. We prove that this Clément operator has second-order approximation properties. For the modified mixed method we show optimal convergence rates of a postprocessed solution under minimal regularity assumptions -- a result not valid for the lowest-order mixed FEM without regularization. Numerical examples conclude this work.

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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.

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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.

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