International Conference on Spectral and High Order Methods
12th - 16th July 2021 | Vienna, Austria
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Please note that all times are shown in the time zone of the conference. The current conference time is: 4th Dec 2022, 07:16:09pm CET
MS 15b: p- and hp-Galerkin methods and approximation of singularities
The mini-symposium focuses on recent developments in p- and hp- Galerkin methods,
most notably for the approximation of singularities. Both theoretical advancements and more
engineering-oriented applications are tackled.
In particular, the mini-symposium will cover the following topics: a priori and a posteriori
error analysis, approximation of isotropic and anisotropic singularities, high-frequency problems,
fractional diffusion, kinetic equations, efficient solvers and geometric/algebraic preconditioners,
boundary elements, and discontinuous Galerkin methods.
12:00pm - 12:30pm
A posteriori error bounds for fully-discrete $hp$-discontinuous Galerkin timestepping methods for parabolic problems
1University of Leicester, United Kingdom; 2National Technical University of Athens, Greece; 3IACM-FORTH, Crete, Greece
We consider fully discrete time-space approximations of abstract parabolic partial differential equations (PDEs) consisting of an $hp$-version discontinuous Galerkin (dG) time stepping scheme in conjunction with (conforming or non-conforming) Galerkin discretizations in space. We derive abstract computable a posteriori error bounds in various norms. We base our methodology for the analysis on a novel space-time reconstruction approach. Our approach is flexible as it works for any type of elliptic error estimator and leaves their choice to the user. It also exhibits mesh-change estimators in a clear and concise way. Lower (efficiency) bounds will also presented for the case of L_2(H^1)-norm a posteriori error estimates, as well as some brief comments on extensions to semilinear PDEs. This will be an overview talk based on results in collaboration Omar Lakkis (Sussex, UK), Charalambos Makridakis (FORTH & Uni, Crete and Sussex, UK) and Thomas Wihler (Bern, Switzerland).
12:30pm - 1:00pm
A-posteriori-steered and adaptive ?-robust multigrid solvers
1TU Wien, Austria; 2Institute of Mathematics, Czech Academy of Sciences, Czech Republic; 3Inria Paris, France
In this work, we study a second-order linear elliptic diffusion problem discretized by the conforming finite element method of arbitrary polynomial degree p ≥ 1. To treat the arising linear system, we propose a geometric multigrid method with zero pre- and one post-smoothing by an overlapping Schwarz (block Jacobi) method. The central feature of our approach consists in introducing optimal step sizes which minimize the algebraic error in the level-wise error correction step of multigrid. This approach leads to an explicit Pythagorean formula for the algebraic error and, importantly, it inherently induces an a posteriori estimator on the energy norm of the algebraic error.
We show the two following results and their equivalence: 1) the solver contracts the algebraic error independently of the polynomial degree p; 2) the estimator represents a two-sided p-robust bound on the algebraic error. The p-robustness results are obtained by carefully applying the results of [J. Schöberl et al., IMA J. Numer. Anal., 28 (2008), pp. 1-24] for one mesh, combined with a multilevel stable decomposition for piecewise affine polynomials of [J. Xu, L. Chen, and R. H. Nochetto, Optimal multilevel methods for H(grad), H(curl), and H(div) systems on graded and unstructured grids, in Multiscale, Nonlinear and Adaptive Approximation, Springer, Berlin, 2009, pp. 599-659]. We consider quasi-uniform or graded bisection simplicial meshes and prove at most linear dependence on the number of mesh levels for minimal H¹-regularity and complete independence for H²-regularity.
Moreover, we develop two adaptive extensions of the algebraic solver. The first extension uses a simple and effective way for the solver to adaptively choose the number of post-smoothing steps, which yields yet improved error reduction. The second extension introduces an adaptive local smoothing strategy which employs additional smoothing only in high-error regions identified thanks to a bulk-chasing criterion and our efficient and localized (by levels/patches of elements) estimator of the algebraic error.
Finally, we present a variety of numerical tests to confirm the p-robust theoretical results and to illustrate the advantages of our adaptive approaches.
1:00pm - 1:30pm
Polynomial-degree-robust a posteriori error estimates for Maxwell's equations
This work proposes novel a posteriori error estimators for Maxwell's equations discretized with Nédélec's finite elements. These estimators are obtained by "broken patchwise flux equilibration" and provide a global upper bound as well as local lower bounds for the discretization error.
The computation of the estimators hinges on solution of local finite element problems on edge patches. These local problems only involve a few degrees of freedom, and can actually be replaced by a sweep through tetrahedra sharing the given edge.
Remarkably, the constants in our upper and lower bounds are independent of the polynomial degree employed, so that the derived estimates are polynomial-degree-robust (or simply p-robust).
I will describe the construction of our estimators, as well as the underlying mathematical foundations. I will also sketch the proofs showing the reliability, efficiency, and p-robustness of the resulting estimates. Finally, I will present numerical experiments that highlight the key features of the proposed error bounds.
1:30pm - 2:00pm
Adaptive deterministic approximation of singular covariance functions for linear elliptic problems under uncertainty
Universität Oldenburg, Germany
When approximating second order moments of elliptic partial differential equations under uncertainty by means of deterministic moment equations, one typically has to deal with two point correlation functions of the right-hand side that have singularities along the diagonal. This poses a challenge as the discretized problem is of twice the dimension of the underlying stochastic elliptic problem and so more sophisticated methods have to be used to resolve the singularities. We propose an adaptive FE method for the solution of deterministic second moment equations in d=2,4 dimensions and show reliability and efficiency estimates of corresponding a posteriori error estimators. Furthermore, numerical experiments validate the theoretical findings.
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