Browsing by Author "Fuhrer, Thomas"

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    A DPG method for shallow shells
    (SPRINGER HEIDELBERG, 2022) Fuhrer, Thomas; Heuer, Norbert; Niemi, Antti H.
    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.
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    A DPG method for the quad-curl problem
    (2023) Fuhrer, Thomas; Herrera, Pablo; Heuer, Norbert
    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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    MULTILEVEL DECOMPOSITIONS AND NORMS FOR NEGATIVE ORDER SOBOLEV SPACES
    (2022) Fuhrer, Thomas
    We consider multilevel decompositions of piecewise constants on simplicial meshes that are stable in H-s for s is an element of (0, 1). Proofs are given in the case of uniformly and locally refined meshes. Our findings can be applied to define local multilevel diagonal preconditioners that lead to bounded condition numbers (independent of the mesh-sizes and levels) and have optimal computational complexity. Furthermore, we discuss multilevel norms based on local (quasi-)projection operators that allow the efficient evaluation of negative order Sobolev norms. Numerical examples and a discussion on several extensions and applications conclude this article.
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    ROBUST DPG TEST SPACES AND FORTIN OPERATORS---THE H1 AND H(div) CASES
    (2024) Fuhrer, Thomas; Heuer, Norbert
    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 H1 and H(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 face bubble functions that are polynomials on faces and decay exponentially in the interior. 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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    Trace operators of the bi-Laplacian and applications
    (OXFORD UNIV PRESS, 2021) Fuhrer, Thomas; Haberl, Alexander; Heuer, Norbert
    We study several trace operators and spaces that are related to the bi-Laplacian. They are motivated by the development of ultraweak formulations for the bi-Laplace equation with homogeneous Dirichlet condition, but are also relevant to describe conformity of mixed approximations. Our aim is to have well-posed (ultraweak) formulations that assume low regularity under the condition of an L-2 right-hand side function. We pursue two ways of defining traces and corresponding integration-by-parts formulas. In one case one obtains a nonclosed space. This can be fixed by switching to the Kirchhoff-Love traces from Fiihrer et al. (2019, An ultraweak formulation of the Kirchhoff-Love plate bending model and DPG approximation. Math. Comp., 88, 1587-1619). Using different combinations of trace operators we obtain two well-posed formulations. For both of them we report on numerical experiments with the discontinuous Petrov-Galerkin method and optimal test functions. In this paper we consider two and three space dimensions. However, with the exception of a given counterexample in an appendix (related to the nonclosedness of a trace space) our analysis applies to any space dimension larger than or equal to two.
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    Trace operators of the bi-Laplacian and applications (vol 41, pg 1031, 2021)
    (OXFORD UNIV PRESS, 2021) Fuhrer, Thomas; Haberl, Alexander; Heuer, Norbert
    © The Author(s) 2020. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.The initial version of this article contained a number of formatting and minor language errors. These were the result of misinterpretation of requested corrections during typesetting. These errors have now been corrected.