### Workshop B6 - Multiresolution and Adaptivity in Numerical PDEs

**Organizers:** Pedro Morin (Universidad Nacional del Litoral, Argentina) - Rob Stevenson (University of Amsterdam, Netherlands) - Christian Kreuzer (Universität Bochum, Germany)

## Talks

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## Mixed adaptive finite element discretization of linear elliptic equations in nondivergence form

### Dietmar Gallistl

### Karlsruher Institut für Technologie, Germany - gallistl@kit.edu

This talk discusses formulations of second-order elliptic partial differential equations in nondivergence form with Cordes coefficients on convex domains as equivalent variational problems. These formulations enable the use of standard finite element techniques for variational problems in subspaces of $H^2$ as well as mixed finite element methods from the context of fluid computations. Besides the immediate quasi-optimal a priori error bounds, the variational setting allows for a posteriori error control with explicit constants and adaptive mesh-refinement. The convergence of an adaptive algorithm is proved. Numerical results on uniform and adaptive meshes are included.

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## Quarklet Frames in Adaptive Numerical Schemes

### Stephan Dahlke

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

This talk is concerned with the design of adaptive numerical schemes for operator equations that are based on subatomic (quarkonial) decompositions. Besides the usual space refinement, in quarkonial decompositions also a poynomial enrichment is included. We construct compactly supported, piecewise polynomial functions whose dilates and translates (quarklets) generate frames for Sobolev spaces. All frame elements except those on the coarsest level have vanishing moment properties. As a consequence, the matrix representation of elliptic operator equations in quarklet frame coordinates is compressible, which is a first important step towards the design of adaptive algorithms.

Joint work with Philipp Keding (Philipps-University of Marburg) and Thorsten Raasch (Johannes Gutenberg-University of Mainz).

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## Adaptive Hierarchical B-Splines: Convergence and Optimality

### Ricardo H Nochetto

### University of Maryland, College Park, USA - rhn@math.umd.edu

We present and analyze an adaptive hierarchical B-splines method, of any (fixed) order and maximal regularity, for linear elliptic equations. We first set the framework for approximation with hierarchical B-splines, and next develop an a posteriori error estimator based on solutions of discrete local problems on stars. We next propose a novel refinement strategy to increase the local resolution of the spaces and derive a contraction property for D\"orfler's marking which implies linear convergence. We also study the complexity of our refinement strategy in terms of marked cells, as well as the optimal relation of energy error vs degrees of freedom for a suitable nonlinear class of functions.

Joint work with Pedro Morin (Universidad Nacional del Litoral, Argentina) and M. Sebastian Pauletti (Universidad Nacional del Litoral, Argentina).

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## Aposteriori error analysis for $hp$-discontinuous Galerkin discretizations of parabolic problems

### Omar Lakkis

### Free University of Bolzano-Bozen, Italy - olakkis@unibz.it

We consider fully discrete space--time approximations of abstract linear parabolic partial differential equations (PDEs) consisting of an $hp$-version discontinuous Galerkin time stepping scheme in conjunction with standard (conforming) Galerkin discretizations in space. We derive computable aposteriori error bounds in the $\operatorname L_\infty(\operatorname L_2)$- and $\operatorname L_2(\operatorname H^1_0)$-type norms based on a novel space-time reconstruction approach.

Joint work with Emmanuil H Georgoulis (University of Leicester, GB, and National Technical University of Athens, GR) and Thomas P Wihler (Universität Bern, CH).

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## Error analysis of a variational multiscale stabilization for convection-dominated diffusion equations in 2d

### Mira Schedensack

### Universität Bonn and Humboldt-Universität zu Berlin, Germany - schedensack@ins.uni-bonn.de

The talk formulates a stabilized quasi-optimal Petrov-Galerkin method for singularly perturbed convection-diffusion problems based on the variational multiscale method. The stabilization is of Petrov-Galerkin type with a standard finite element trial space and a problem-dependent test space based on precomputed fine-scale correctors. The exponential decay of these correctors and their localisation to local patch problems, which depend on the direction of the velocity field and the singular perturbation parameter, is rigorously justified. Under moderate assumptions, this stabilization guarantees stability and quasi-optimal rate of convergence for arbitrary mesh Peclet numbers on fairly coarse meshes at the cost of additional inter-element communication.

Joint work with Guanglian Li (Universität Bonn, Germany) and Daniel Peterseim (Universität Bonn, Germany).

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## Improved regularity estimates for solutions to the $p$-Poisson equation

### Markus Weimar

### University of Siegen, Germany - weimar@mathematik.uni-siegen.de

It is well-known that the rate of convergence of numerical schemes which aim to approximate solutions to operator equations is closely related to the maximal regularity of these solutions in certain scales of smoothness spaces of Sobolev and Besov type. For linear elliptic PDEs on Lipschitz domains, a lot of results in this direction already exist. In contrast, it seems that not too much is known for nonlinear problems.

In this talk, we are concerned with the $p$-Laplace operator which has a similar model character for quasi-linear equations as the ordinary Laplacian for linear problems. It finds applications in models, e.g., for turbulent flows of a gas in porous media, radiation of heat, as well as in non-Newtonian fluid theory. We discuss a couple of local regularity estimates for the gradient of the unknown solutions. These assertions are then used to derive global smoothness properties by means of wavelet-based proof techniques. Finally, the presented results imply that adaptive algorithms are able to outperform (at least asymptotically) their commonly used counterparts based on uniform refinement.

The material extends assertions which were obtained earlier in joint work with S. Dahlke, L. Diening, C. Hartmann, and B. Scharf (Nonlinear Anal., 130:298-329, 2016).

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## Computation of local and quasi-local effective diffusion tensors in elliptic homogenization

### Daniel Peterseim

### University of Bonn, Germany - peterseim@ins.uni-bonn.de

The talk discusses a re-interpretation of existing multiscale methods by means of a discrete integral operator acting on standard finite element spaces. The exponential decay of the involved integral kernel motivates the use of a diagonal approximation and, hence, a localized piecewise constant effective coefficient. This local model turns out to be appropriate when the localized coefficient satisfies a certain homogenization criterion, which can be verified a posteriori. An a priori error analysis of the local model is presented and illustrated in numerical experiments.

Reference: D. Gallistl and D. Peterseim: Computation of local and quasi-local effective diffusion tensors in elliptic homogenization. ArXiv e-print 1608.02092, 2016.

Joint work with Dietmar Gallistl (Karlsruhe Institute of Technology, Germany).

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## Adaptive Finite Element Methods for Unresolved Diffusion Coefficients

### Andrea Bonito

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

Elliptic partial differential equations (PDEs) with discontinuous diffusion coefficients occur in application domains such as diffusions through porous media, electro-magnetic field propagation on heterogeneous media, and diffusion processes on rough surfaces.

In most applications, the discontinuities do not lie on the boundaries of the cells in the initial triangulation. Rather, the discontinuities occur on curves, surfaces, or manifolds, and could even be unknown beforehand. One of the obstacles to treating such discontinuity problems is that the usual perturbation theory for elliptic PDEs assumes bounds for the distortion of the coefficients in the $L_\infty$ norm and this in turn requires that the discontinuities are matched exactly when the coefficients are approximated.

We present a new approach based on distortion of the coefficients in an $L_q$ norm with $q<\infty$, which therefore does not require the exact matching of the discontinuities. We then use this new distortion theory to formulate new adaptive finite element methods (AFEMs) for such discontinuity problems. We show that such AFEMs are optimal in the sense of distortion versus number of computations, and report insightful numerical results supporting our analysis.

If time permits, we will proceed further and discuss how this perturbation theory can be used in turn to derive stability estimates for parameter recovery processes in diffusion problems.

Refs:

A. Bonito, R.A. DeVore, and R.H. Nochetto: Adaptive Finite Element Methods for Elliptic Problems with Discontinuous Coefficients.

A. Bonito, A. Cohen, R.A. DeVore, G. Petrova, and G. Welper: Diffusion Coefficients Estimation for Elliptic Partial Differential Equations.

Joint work with Ronald A. DeVore (Texas A&M University, USA) and Ricardo H. Nochetto (University of Maryland, USA).

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## Oscillation in a posteriori error estimation

### Andreas Veeser

### Università degli Studi di Milano, Italy - andreas.veeser@unimi.it

In a posteriori error estimation, one devises computationally accessible estimators for the error of an approximate PDE solution. Such estimators allow evaluating the approximation quality and are also used to guide adaptive mesh refinement.

In the available approaches to a posteriori error estimation, the estimators are equivalent to the approximation error, up to so-called oscillation terms. These terms vanish for data resolved by the underlying mesh, but they can be arbitrarily bigger than the error. Oscillation was believed to be exclusively the price of (a better) computability and to converge at least as fast as the error. A remarkable example of Cohen, DeVore and Nochetto (2012) shows that both beliefs are wrong for the oscillation terms in the available approaches.

This talk presents a new approach to a posteriori error estimation and applications. In contrast to preceding ones, the arising, new oscillation terms are bounded by the error. The new twist is an interpolation operator that splits the residual into an approximate residual with oscillation-free data and second part concerning the not yet resolved data.

Joint work with Christian Kreuzer (Ruhr-Universität Bochum, Germany).

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## Near-Best Approximation on Adaptive Partitions

### Peter Binev

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

We consider adaptive approximation of a function on a given domain $\Omega$ using piecewise polynomial functions. To study this approximation, the partitioning of $\Omega$ is related to building a binary tree $T$ whose leaves represent the elements of the partition. The problem of approximating a function is presented then as the process of identifying a tree $T$ and the related approximating polynomials $P(\Delta)$ for each of its leaves $\Delta$. We consider two cases of adaptivity: the h-adaptivity in which the polynomials $P(\Delta)$ are of the same order and the hp-adaptivity in which the orders of $P(\Delta)$ may vary for different $\Delta$-s. We use error functionals to drive the adaptive process and prove that the presented algorithms provide a near-best approximation in both cases.

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## $H^1$-stability of the $L^2$-projection and applications to adaptive methods

### Fernando Gaspoz

### University of Stuttgart, Germany - fernando.gaspoz@ians.uni-stuttgart.de

The $L^2$-projection onto discrete spaces plays an essential role in the analysis of finite element discretizations. On uniform grids $H^1$-stability of the $L^2$-projection can easily be deduced by an inverse estimate. This simple proof hinges on the fact that the minimal mesh-size is comparable to the maximal mesh-size. We provide a technique that sidestep this restriction and prove the stability in $H^1$ of the $L^2$-projection for piecewise continuous Finite Element Spaces for a class of adaptive meshes. We also present some new applications of this estimate.

Joint work with Claus-Justus Heine (University of Stuttgart, Germany) and Kunibert Siebert (University of Stuttgart, Germany).

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## Adaptive finite element methods for $C^2$ surfaces

### Alan Demlow

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

FEM for elliptic PDE on smooth surfaces involve approximating both the surface itself (leading to a "geometric" consistency error) and the PDE (leading to a standard Galerkin error). The behavior of the geometric error depends heavily on the choice of surface representation. Parametric representations based on mappings from Euclidean domains are more flexible and generally easier to implement, but lead to a lower-order (larger) geometric error. Implicit representations based on a closest-point projection are less flexible because they require a $C^2$ surface and because the closest-point projection is not usually analytically computable. However, the geometric error derived from such representations is of higher order due to special properties of the closest-point projection. Because existing estimators based on the closest-point projection require actually computing it, non-computability of the closest-point projection becomes critical when carrying out adaptive computations. In this talk we merge these two perspectives by constructing estimators which preserve many of the best properties of closest-point and parametric estimators and discuss behavior of AFEM based on our new estimators.

Joint work with Andrea Bonito (Texas A&M University).

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## On the error control for fully discrete approximations of the time-dependent Stokes equation.

### Fotini Karakatsani

### University of Chester, UK - f.karakatsani@chester.ac.uk

We consider fully discrete finite element approximations to the time-dependent Stokes system. The space discretization is based on popular stable spaces, including Crouzeix-Raviart and Taylor-Hood finite element methods. Implicit Euler is applied for the time discretization. The finite element spaces are allowed to change with time steps and the projection steps include alternatives that are hoped to cope with possible numerical artifices and the loss of the discrete incompressibility of the schemes. The final estimates are of optimal order in $L^\infty (L^2) $ for the velocity error.

Joint work with Eberhard B\"ansch (University of Erlangen, Germany), Charalambos Makridakis (University of Sussex, UK).

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## A posteriori error estimates in $L^2(H^1)\cap H^1(H^{-1})$ and $L^2(H^1)$-norms for high-order discretizations of parabolic problems

### Iain Smears

### Inria Paris, France - iain.smears@inria.fr

We present a posteriori error estimates in parabolic energy norms for space-time discretisations based on arbitrary-order conforming FEM in space and arbitrary-order discontinuous Galerkin methods in time. Using the heat equation as a model problem, we first consider a posteriori error estimates in a norm of $L^2(H^1)\cap H^1(H^{-1})$-type that is extended to the nonconforming discrete space. We construct a flux equilibration by solving at each time-step some local mixed finite element problems posed on the patches of the mesh, which yields estimators with guaranteed upper bounds on the error, and locally space-time efficient lower bounds with respect to this extended norm. Furthermore, the efficiency constants are robust with respect to the discretisation parameters, including the polynomial degrees in both space and time, and also with respect to refinement and coarsening between time-steps, thereby removing the need for the transition conditions required in earlier works. In the last part of the talk, we consider the same flux equilibration in the context of $L^2(H^1)$-norm a posteriori error estimation, where we show that in the practically relevant situation where the time-step size $\tau \gtrsim h^2$ the mesh-size, then the spatial part of the estimators have the additional feature of being locally efficient with respect to the $L^2(H^1)$-norm of the error plus the temporal jumps. Our analysis of efficiency in $L^2(H^1)$-norms thus removes the more severe time-step and mesh size conditions required previously in the literature.

Joint work with Alexandre Ern (ENPC, France) and Martin Vohralik (Inria Paris, France).

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## Rate optimal adaptivity for non-symmetric FEM/BEM coupling

### Michael Feischl

### UNSW Sydney, Australia - m.feischl@unsw.edu.au

We develop a framework which allows us to prove the essential general quasi-orthogonality for the non-symmetric Johnson-Nédélec finite element/boundary element coupling. General quasi-orthogonality was first proposed in [Carstensen, Feischl, Page, Praetorius 2014] as a necessary ingredient of optimality proofs and is the major difficulty on the way to prove rate optimal convergence of adaptive algorithms for many strongly non-symmetric problems. The proof exploits a new connection between the general quasi-orthogonality and $LU$-factorization of infinite matrices. We then derive that a standard adaptive algorithm for the Johnson-Nédélec coupling converges with optimal rates. The developed techniques are fairly general and can most likely be applied to other problems like Stokes equation.

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## Guaranteed and robust a posteriori bounds for Laplace eigenvalues and eigenvectors

### Martin Vohralik

### Inria Paris, France - martin.vohralik@inria.fr

We derive a posteriori error estimates for numerical approximation of the Laplace eigenvalue problem with a homogeneous Dirichlet boundary condition. In particular, upper and lower bounds for an arbitrary simple eigenvalue are given. These bounds are guaranteed, fully computable, and converge with optimal speed to the given exact eigenvalue, under a separation condition from the surrounding smaller and larger exact eigenvalues that we can check in practice. Guaranteed, fully computable, optimally convergent, and polynomial-degree robust bounds on the energy error in the approximation of the associated eigenvector are derived as well, under the same hypotheses. Remarkably, there appears no unknown (solution-, regularity-, or polynomial-degree-dependent) constant in our theory. Inexact algebraic solvers are taken into account, so that the estimates are valid on each iteration and can serve for design of adaptive stopping criteria for eigenvalue solvers. The framework can be applied to conforming, nonconforming, discontinuous Galerkin, and mixed finite element approximations of arbitrary polynomial degree. We illustrate it numerically on a set of test problems.

Joint work with Eric Cances (Ecole des Ponts, France), Genevieve Dusson (University Paris 6, France), Yvon Maday (University Paris 6, France) and Benjamin Stamm (RWTH Aachen, Germany).

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## Reliable computations and self adapted methods in nonlinear phenomena

### Charalambos Makridakis

### IACM-FORTH Crete and Univ. of Sussex - c.makridakis@sussex.ac.uk

The computation of singular phenomena (shocks, defects, dislocations, interfaces, cracks) arises in many complex systems. For computing such phenomena, it is natural to seek methods that are able to detect them and to devote the necessary computational recourses to their accurate resolution. At the same time, we would like to have mathematical guarantees that our computational methods approximate physically relevant solutions. Our purpose in this talk is to review results and discuss related computational challenges for such nonlinear problems modelled either by PDEs or are a result of Micro / Macro adaptive modelling methods.

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## Optimal convergence rates for adaptive FEM for compactly perturbed elliptic problems

### Dirk Praetorius

### TU Wien, Austria - dirk.praetorius@tuwien.ac.at

We consider adaptive FEM for problems, where the corresponding bilinear form is symmetric and elliptic up to some compact perturbation. We prove that adaptive mesh-refinement is capable of overcoming the preasymptotic behavior and eventually leads to convergence with optimal algebraic rates. As an important consequence of our analysis, one does not have to deal with the a priori assumption that the underlying meshes are sufficiently fine. In particular, our analysis covers adaptive mesh-refinement for the Helmholtz equation from where our interest originated.

Joint work with Alex Bespalov (University of Birmingham, UK) and Alexander Haberl (TU Wien, Austria).

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## Local estimates for the discrete (p-)harmonic functions for fully adaptive meshes

### Lars Diening

### University of Bielefeld, Germany - lars.diening@uni-bielefeld.de

It is well known that harmonic functions and p-harmonic functions have higher interior regularity. In 1957 De~Giorgi introduced a new technique that allows for example to estimate the maximum of the solution on a ball by an mean integral of the solution on an enlarged ball. A similar result holds for $p$-harmonic functions. The proof is based on subtle Cacciopoli estimates using truncation operators. In this talk we present similar estimates for discretely harmonic and $p$-harmonic functions. Our solutions can be scalar valued as well as vector valued, which makes a big difference for~$p$-harmonic functions.

Various results already exist in this direction for harmonic functions (e.g. Schatz-Wahlbin '95). However, the main obstacle in this direction even in the linear case is adaptivity. All of the results obtained so far, require that the mesh size does not vary too much locally. Our approach differs in such that we allow for arbitrary highly graded meshes (still shape regular). However, our approach uses certain properties of the Lagrange basis functions. This restrict our approach at the moment to acute meshes and linear elements. The proof of our result is based on a discretized version of the De~Giorgi technique.

Joint work with Toni Scharle (Oxford University, Great Britain).

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## Transport and Adaptivity

### Wolfgang Dahmen

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

Linear transport equations arise as limits of convection diffusion equations and form core constituents of kinetic models of Boltzmann type. We begin with indicating the importance of stable variational formulations of transport equations in such contexts. We discuss, in particular, the relevance of efficient and reliable a posteriori error bounds that have not been available so far. We then present some core ingredients of deriving such error bounds. We highlight essential distinctions from elliptic type problems which are mainly due to the anisotropic structure of the relevant function spaces arising in connection with transport problems. In particular, due to the fact that the a posteriori error indicators do not depend explicitly on mesh size parameters, it becomes much more involved to show that refinements, based on typical bulk-criteria, result in a fixed error reduction rate. We indicate some of the key ingredients and conclude with a short outlook on further consequences.

Joint work with Felix Gruber (RWTH Aachen), Olga Mula (Paris Dauphine University) and Rob Stevenson (University of Amsterdam).