Central Workshop on Adaptive Finite Element Methods
TU Wien, Friday 10.11.2017
 Seminarraum SEM DA grün 03 C (green section, 3rd floor)
Schedule
time  speaker  title 

10:00  11:00  Dirk Praetorius (TU Wien) 
AFEM with inhomogeneous Dirichlet data 
11:00  11:30  Coffee break  
11:30  12:30  Carsten Carstensen (HU Berlin) 
Optimal convergence rates for adaptive lowestorder discontinuous PetrovGalerkin schemes 
12:30  14:00  Lunch break  
14:00  14:45  Philipp Bringmann (HU Berlin) 
Rate optimal adaptive leastsquares finite element scheme for the Stokes equations 
14:45  15:30  Alexander Haberl (TU Wien) 
Rate optimal adaptive FEM with inexact solver for nonlinear operators 
15:30  16:00  Coffee break  
16:00  17:00  Joscha Gedicke (Uni Wien) 
Residualbased a posteriori error analysis for symmetric mixed ArnoldWinther FEM 
Abstracts
Philipp Bringmann (HU Berlin) 

Rate optimal adaptive leastsquares finite element scheme for the Stokes equations 
The talk concerns the first adaptive leastsquares finite element method (LSFEM) for the Stokes equations with optimal convergence rates based on the newest vertex bisection with lowestorder RaviartThomas and conforming P1 discrete spaces for the divergence leastsquares formulation in 2D. Although the leastsquares functional is a reliable and efficient error estimator, the novel refinement indicator stems from an alternative explicit residualbased a posteriori error control with exact solve. Particular interest is on the treatment of the data approximation error which requires a separate marking strategy. The paper proves linear convergence in terms of the levels and optimal convergence rates in terms of the number of unknowns relative to the notion of a nonlinear approximation class. It extends and generalizes the approach of Carstensen and Park (SIAM J. Numer. Anal. 53:4362 2015) from the Poisson model problem to the Stokes equations with inhomogeneous Dirichlet boundary conditions. Further generalizations to 3D linear elasticity with inhomogeneous Neumann conditions (joint work with Carsten Carstensen and Gerhard Starke) and higher polynomial degrees for an hadaptive LSFEM (joint work with Carsten Carstensen) are possible. Reference:

Carsten Carstensen (HU Berlin) 
Optimal convergence rates for adaptive lowestorder discontinuous PetrovGalerkin schemes 
The discontinuous PetrovGalerkin methodology enjoys a builtin a posteriori error control in some computable residual term plus data approximation terms. This talk advocates an alternative error estimator, which is globally equivalent, but allows for the proof of the axioms of adaptivity and so guarantees optimal convergence rates of the associated adaptive algorithm. The talk exemplifies the analysis for the Poisson model problem with a righthand side f in L2 in the polyhedral domain simultaneously for the four lowestorder discontinuous PetrovGalerkin schemes. Those are rewritten in terms of the firstorder nonconforming CrouzeixRaviart functions, with respect to a shaperegular triangulation T into simplices, some projection Q and the Galerkin projection G onto the conforming P1 finite element subspace. The novel error estimator consists of the expected volume contributions and the jump terms of the piecewise gradient of the CrouzeixRaviart solution across the sides of any simplex. The estimator exclusively involves the CrouzeixRaviart solution and seemingly ignores the conforming contribution, but surprisingly also controls the total error term. The optimal convergence rates rely on standard arguments for stability and reduction, while the discrete reliability involves an additional term and eventually enforces the additional condition of a sufficiently small initial meshsize for optimal convergence rates. The presentation is on ongoing joint work with Friederike Hellwig. Related References:

Joscha Gedicke (Uni Wien) 
Residualbased a posteriori error analysis for symmetric mixed ArnoldWinther FEM 
This talk introduces an explicit residualbased a posteriori error analysis for the symmetric mixed finite element method in linear elasticity after ArnoldWinther with pointwise symmetric and divconforming stress approximation. Opposed to a previous publication, the residualbased a posteriori error estimator of this talk is reliable and efficient and truly explicit in that it solely depends on the symmetric stress and does neither need any additional information of some skew symmetric part of the gradient nor any efficient approximation thereof. Hence it is straightforward to implement an adaptive meshrefining algorithm obligatory in practical computations. Numerical experiments verify the proven reliability and efficiency of the new a posteriori error estimator and illustrate the improved convergence rate in comparison to uniform meshrefining. A higher convergence rate for piecewise affine data is observed in the stress error and reproduced in nonsmooth situations by the adaptive meshrefining strategy. This is joint work with Carsten Carstensen and Dietmar Gallistl. 
Alexander Haberl (TU Wien) 
Rate optimal adaptive FEM with inexact solver for nonlinear operators 
Analyzing an algorithm from [Congreve—Wihler, JCAM 311, 2017, we prove convergence with optimal algebraic rates for an adaptive finite element method for nonlinear equations with strongly monotone operator. The analysis also includes the iterative and inexact solution of the arising nonlinear systems by means of the Picard iteration. Using nested iteration, we prove that the number of Picard iterations is generically bounded, and the overall computational cost is (almost) optimal. The talk is based on joint work with Gregor Gantner, Dirk Praetorius, and Bernhard Stiftner. Reference:

Dirk Praetorius (TU Wien) 
AFEM with inhomogeneous Dirichlet data 
We consider the adaptive finite element method for general secondorder linear elliptic PDEs. If the PDE is supplemented by inhomogeneous Dirichlet conditions, the adaptive algorithm has to steers the discretization of the given Dirichlet data as well as the resolution of the possible singularities of the (unknown) solution. We discuss possible strategies as well as available a posteriori error estimators. The focus of the talk is on related rate optimality results. Numerical experiments and open problems conclude the talk. Related References:
