# Session A5 - Multiresolution and Adaptivity in Numerical PDEs

Organizers: Annalisa Buffa (IMATI, Italy) - Angela Kunoth (University of Cologne, Germany) - Pedro Morín (Universidad Nacional del Litoral, Argentina)

Workshop website

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## Talks

December 11, 14:30 ~ 15:10 - Room B22

## A posteriori error estimation and adaptivity in the maximum norm

### Texas A&M University, USA   -   demlow@math.tamu.edu

In this talk I will discuss recent progress in constructing adaptive algorithms for controlling errors in the maximum norm. The main focus of the talk will be the construction of robust a posteriori error estimators for singularly perturbed elliptic reaction-diffusion problems. Time permitting, I may also discuss the sharpness of logarithmic factors that typically arise in maximum norm a posteriori estimates.

Joint work with Natalia Kopteva (University of Limerick).

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December 11, 15:35 ~ 16:25 - Room B22

## Weak convergence analysis for stochastic evolution PDEs

### Chalmers University of Technology and University of Gothenburg, Sweden   -   stig@chalmers.se

I will review the literature on the weak convergence analysis of numerical methods for stochastic evolution problems driven by noise. Then I will present a new method of proof, which is based on refined Sobolev-Malliavin spaces from the Malliavin calculus. It does not rely on the use of the Kolmogorov equation or the Ito formula and is therefore applicable also to non-Markovian equations, where these are not available. We use it to prove weak convergence of fully discrete approximations of the solution of the semilinear stochastic parabolic evolution equation with additive noise as well as a semilinear stochastic Volterra integro-differential equation.

Joint work with Adam Andersson (Chalmers University of Technology and University of Gothenburg, Sweden), Mihaly Kovacs (University of Otago, New Zealand) and Raphael Kruse (TU Berlin, Germany).

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December 11, 17:00 ~ 17:20 - Room B22

## Existence of $p$-Moments for the Weak Space-Time Heat Equation with Random Coefficients and Stability of its Petrov-Galerkin Discretization

### University of Cologne, Germany   -   cmollet@math.uni-koeln.de

We consider the heat equation in a weak space-time formulation with random right hand side and random spatial operator. The existence and uniqueness of a solution can be proven by the Banach-Necas-Babuska theorem. In this course we allow the spatial operator $A$ to have lower and upper bounds depending on a stochastic parameter $\omega$, i.e., we consider random variables $A_{\rm \min }(\omega)$ and $A_{\rm max}(\omega)$ as lower and upper bounds. The $L_p$-regularity of the solution and its connection to the random variables bounding $A$ can be proven. A similar approach is applied to a full space-time Petrov-Galerkin discretization. The stability of such an approach requires a lower bound for the discrete inf-sup condition independent of the grid spacing. Using similar ideas, we can prove stability when allowing a finer discretization for the test space than for the solution space.

Joint work with Stig Larsson (Chalmers University of Technology) and Matteo Molteni (Chalmers University of Technology).

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December 11, 17:30 ~ 17:50 - Room B22

## Regularity of boundary integral equations in Besov-type spaces based on wavelet expansions

### Philipps-University Marburg, Germany   -   weimar@mathematik.uni-marburg.de

We study regularity properties of solutions to operator equations on patchwise smooth manifolds $\partial\Omega$ such as, e.g., boundaries of polyhedral domains $\Omega \subset \mathbb{R}^3$. Using suitable biorthogonal wavelet bases $\Psi$, we introduce a new class of Besov-type spaces $B_{\Psi,q}^\alpha(L_p(\partial \Omega))$ of functions $u\colon\partial\Omega\rightarrow\mathbb{C}$. Special attention is paid on the rate of convergence for best $n$-term wavelet approximation to functions in these scales since this determines the performance of adaptive numerical schemes. We show embeddings of (weighted) Sobolev spaces on $\partial\Omega$ into $B_{\Psi,\tau}^\alpha(L_\tau(\partial \Omega))$, $1/\tau=\alpha/2 + 1/2$, which lead us to regularity assertions for the equations under consideration. Finally, we apply our results to a boundary integral equation of the second kind which arises from the double layer ansatz for Dirichlet problems for Laplace's equation in $\Omega$. The talk is based on two recent papers which arose from the DFG-Project ''BIOTOP: Adaptive Wavelet and Frame Techniques for Acoustic BEM'' (DA 360/19-1):

Dahlke, S. and Weimar, M.: Besov regularity for operator equations on patchwise smooth manifolds. Preprint 2013-03, Fachbereich Mathematik und Informatik, Philipps-Universität Marburg. To appear in J. Found. Comput. Math.

Weimar, M.: Almost diagonal matrices and Besov-type spaces based on wavelet expansions. Preprint 2014-06, Fachbereich Mathematik und Informatik, Philipps-Universität Marburg. Submitted.

Joint work with Stephan Dahlke (Philipps-University Marburg, Germany).

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December 12, 14:30 ~ 15:10 - Room B22

### University of South Carolina, United States   -   binev@math.sc.edu

The hp-adaptive numerical methods for PDEs combine the domain decomposition with assignment of degrees of freedom at each element of a particular refinement. The main objective of this talk is to introduce a framework that streamlines the process of making adaptive decisions.

We consider domain partitioning based on a fixed binary refinement scheme and a coarse-to-fine routine for making adaptive decisions about the elements to be split and the polynomial orders to be assigned. The problem of finding near-optimal results is managed by using greedy algorithms on binary trees and a modification of the local errors that take into account the local complexity of the adaptive approximation. We prove that the algorithm provides near-best approximation to any given function.

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December 12, 15:30 ~ 16:10 - Room B22

## convergence and optimality of adaptive boundary element methods

### Vienna University of Technology, Austria   -   melenk@tuwien.ac.at

We consider the $h$-version of the boundary element method (BEM) in 2D and 3D on shape regular meshes. We show convergence and prove quasi-optimality of an adaptive BEM (ABEM), taking Symm's integral equation as our model problem for a first kind integral equation. Optimality here means that the algorithm realizes the optimal rate achievable for solutions for an approximation class that is characterized by the best possible decay rate achievable for the error indicator under the mesh refinements allowed (here: newest vertex bisection). The error indicators that drive the adaptive algorithm are of residual type and hark back to \cite{carstensen-stephan95,carstensen-maischak-stephan01}. For the FEM on shape regular meshes, similar convergence and optimality results are available, (Stevenson 2007; Cascon, Kreuzer, Nochetto, Siebert 2008). The BEM setting is more involved and requires different mathematical tools since the operators and pertinent norms are non-local. This mandates in particular the use of non-standard inverse estimates for integral operators, which we present in this talk. We will also discuss extensions of the algorithm to account for data approximations and applications to FEM-BEM coupling.

Joint work with Michael Feischl (Vienna University of Technology, Austria), Michael Karkulik (Pontificia Universidad Catolica de Chile, Chile) and Dirk Praetorius (Vienna University of Technology, Austria).

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December 12, 17:05 ~ 17:55 - Room B22

## Adaptive low-rank tensor approximation for high dimensional operator equations

### RWTH Aachen, Germany   -   dahmen@igpm.rwth-aachen.de

Problems in high spatial dimensions are typically subject to the "curse of dimensionality" which roughly means that the computational work, needed to approximate a given function within a desired target accuracy, increases exponentially in the spatial dimension. A possible remedy is to seek problem dependent dictionaries with respect to which the function possesses sparse approximations. Employing linear combinations of particularly adapted rank-one tensors falls into this category. In this talk we highlight some recent developments centering on the adaptive solution of high dimensional operator equations in terms of stable tensor expansions. Some new concepts related to tensor contractions, tensor recompression, coarsening, and rescaling operators are outlined. Some essential issues are addressed that arise in the convergence and complexity analysis but have been largely ignored when working in a fully discrete setting. In particular, when dealing with high-dimensional diffusion problems, a central obstruction is related to the spectral properties of the underlying operator which is an isomorphism only when acting between spaces that are not endowed with tensor product norms. The theoretical results are illustrated by numerical experiments.

Joint work with Markus Bachmayr (RWTH-Aachen, Germany).

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December 12, 18:00 ~ 18:40 - Room B22

## Polynomial-degree-robust a posteriori estimates in a unified setting

### INRIA Paris-Rocquencrourt, France   -   martin.vohralik@inria.fr

We present equilibrated flux a posteriori error estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed finite element discretizations of the two-dimensional Poisson problem. Relying on the equilibration by mixed finite element solution of patchwise Neumann problems, the estimates are guaranteed, locally computable, locally efficient, and robust with respect to polynomial degree. Maximal local overestimation is guaranteed as well. Numerical experiments suggest asymptotic exactness for the incomplete interior penalty discontinuous Galerkin scheme.

Joint work with Alexandre Ern (CERMICS, Universite Paris-Est, France).

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December 13, 14:30 ~ 15:10 - Room B23

## Adaptive Wavelet Boundary Element Methods

### University of Basel, Switzerland   -   helmut.harbrecht@unibas.ch

This talk is concerned with developing numerical techniques for the adaptive application of global operators of potential type in wavelet coordinates. This is a core ingredient for a new type of adaptive solvers that has so far been explored primarily for partial differential equations. We shall show how to realize asymptotically optimal complexity in the present context of global operators. Asymptotically optimal means here that any target accuracy can be achieved at a computational expense that stays proportional to the number of degrees of freedom (within the setting determined by an underlying wavelet basis) that would ideally be necessary for realizing that target accuracy if full knowledge about the unknown solution were given. The theoretical findings are supported and quantified by numerical experiments.

Joint work with Manuela Utzinger (University of Basel).

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December 13, 15:30 ~ 16:10 - Room B23

## A convergent adaptive scheme for hierarchical isogeometric methods

### INdAM c/o University of Florence, Italy   -   carlotta.giannelli@unifi.it

Adaptive finite element methods (AFEM) usually rely on an iterative procedure that consists of four fundamental modules. For any iterative step, the adaptive loop starts with the approximation of the solution with respect to the current computational mesh. A posteriori error estimates are then computed in terms of local indicators associated to any single mesh element. Subsequently, a marking strategy selects the elements with higher values of the local error indicator. Finally, a refinement procedure constructs the refined mesh starting from the set of marked elements. In general, the refinement procedure identifies the mesh with an increased level of resolution for the next iteration by refining not only the marked elements, but also a suitable set of elements in their neighbourhood. This may allow to guarantee certain properties of the resulting mesh that preserve the error estimates previously computed. In particular, specific bounds for the number of the non-zero basis functions on any mesh element plays a key role for the development of an adaptivity theory. In order to extend recent results obtained in the AFEM context to the isogeometric setting, we rely on local refinement techniques based on adaptive spline spaces. In particular, by exploiting the truncated basis for hierarchical B-spline spaces together with suitable mesh configurations, we will provide simple residual-type error estimates and prove the convergence of the adaptive procedure.

Joint work with Annalisa Buffa (IMATI “E. Magenes” - CNR, Italy).

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December 13, 17:00 ~ 17:40 - Room B23

## Adaptive methods for nonlinear problems: PDEs and Multiscale Modelling

### University of Sussex, UK   -   c.makridakis@sussex.ak.uk

The computation of singular phenomena (shocks, defects, dislocations,