13:30
MS3: Adaptivity, Regularity and Fast Approximation (Part 2)
Chair: Markus Weimar
13:30
25 mins
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Adaptive methods for solving parabolic evolution equations.
Rob Stevenson
Abstract: Recently we constructed an optimally converging adaptive method for solving parabolic evolution equations. This method was based on a space-time formulation of the problem as a first order system least squares system, and the application of tensor product wavelet bases. Although uniformly bounded in the problem size, the computational work per unknown turned out to be relatively high.
Aiming at better quantitative results, in this talk we describe our current efforts towards achieving similar theoretical results concerning optimality using finite element discretizations in space. We discuss different variational formulations, inf-sup stability, optimal preconditioners, a posteriori estimators, and adaptivity.
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13:55
25 mins
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Besov Regularity of Solutions to Linear and Nonlinear Parabolic PDEs
Stephan Dahlke, Cornelia Schneider
Abstract: We will be concerned with regularity estimates for solutions to linear and nonlinear parabolic partial differential equations on nonsmooth domains. In particular, we will discuss the smoothness in the specific scale $B^s_{\tau \tau}, 1/{\tau}=s/d +1/p$ of Besov spaces. The regularity in these spaces determines the approximation order that can be achieved by adaptive and other nonlinear approximation schemes. We will show that for all cases under consideration the Besov regularity is high enough to justify the use of adaptive algorithms.
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14:20
25 mins
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Rate optimal adaptive FEM with inexact solver for nonlinear operators
Gregor Gantner, Alexander Haberl, Dirk Praetorius, Stefan Schimanko
Abstract: We present our recent work [Gantner et al., IMA J Numer Anal 38, 2018], where we prove convergence with optimal algebraic rates for an adaptive finite element method for nonlinear equations with strongly monotone operator. We consider an algorithm proposed by [Congreve et al., J Comp Appl Math 311, 2017]. Unlike prior works, e.g., [Carstensen et al., Comp Math Appl 67, 2014], our analysis also includes the iterative and inexact solution of the arising nonlinear systems by means of the Picard iteration. Using nested iteration, we prove, in particular, that the number of Picard iterations is uniformly bounded in generic cases. Finally, we aim to discuss that the overall computational cost is optimal. Numerical experiments confirm the theoretical results.
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14:45
25 mins
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Higher order regularity shifts for the p-Poisson problem
Anna Kh. Balci, Lars Diening, Markus Weimar
Abstract: We discuss new local regularity estimates related to the $p$-Poisson equation $-\mathrm{div}(A(\nabla u))=-\mathrm{div} F$ for $p>2$. In the planar case $d=2$ we are able to transfer local interior Besov and Triebel-Lizorkin regularity up to first order derivatives from the forcing term $F$ to the flux $A(\nabla u)=\left|\nabla u\right|^{p-2}\nabla u$. In case of higher dimensions or systems we have a smallness restriction on the corresponding smoothness parameter. Apart from that, our results hold for all reasonable parameter constellations related to weak solutions $u\in W^{1,p}(\Omega)$ including quasi-Banach cases with applications to adaptive finite element analysis.
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