Central Workshop on Adaptive Finite Element Methods

The talk concerns the first adaptive least-squares finite element method (LS-FEM) for the Stokes equations with optimal convergence rates based on the newest vertex bisection with lowest-order Raviart-Thomas and conforming P1 discrete spaces for the divergence least-squares formulation in 2D. Although the least-squares functional is a reliable and efficient error estimator, the novel refinement indicator stems from an alternative explicit residual-based 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 non-linear approximation class. It extends and generalizes the approach of Carstensen and Park (SIAM J. Numer. Anal. 53:43-62 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 h-adaptive LS-FEM (joint work with Carsten Carstensen) are possible.

Reference:

• P. Bringmann, C. Carstensen: An adaptive least-squares FEM for the Stokes equations with optimal convergence rates, Numerische Mathematik, 135 (2017), 459-492, DOI: 10.1007/s00211-016-0806-1

The discontinuous Petrov-Galerkin methodology enjoys a built-in 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 right-hand side f in L2 in the polyhedral domain simultaneously for the four lowest-order discontinuous Petrov-Galerkin schemes. Those are rewritten in terms of the first-order nonconforming Crouzeix-Raviart functions, with respect to a shape-regular 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 Crouzeix-Raviart solution across the sides of any simplex. The estimator exclusively involves the Crouzeix-Raviart 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 mesh-size for optimal convergence rates.

The presentation is on ongoing joint work with Friederike Hellwig.

Related References:

• C. Carstensen, L. Demkowicz, and J. Gopalakrishnan, A Posteriori Error Control for DPG Methods, SIAM J. Numer. Anal., 52 (2014), pp. 1335-1353, doi:10.1137/130924913.
• C. Carstensen, L. Demkowicz, and J. Gopalakrishnan, Breaking spaces and forms for the DPG method and applications including Maxwell equations, Comput. Math. Appl., 72 (2016), pp. 494-522, doi:10.1016/j.camwa.2016.05.004.
• C. Carstensen and H. Rabus. Axioms of adaptivity with separate marking for data resolution. SIAM Journal on Numerical Analysis, 2017, accepted, arXiv:1606.02165.
• C. Carstensen, M. Feischl, M. Page, and D. Praetorius, Axioms of adaptivity, Comput.Math. Appl., 67 (2014), pp. 1195-1253, doi:10.1016/j.camwa.2013.12.003.
• C. Carstensen, D. Gallistl, F. Hellwig, and L. Weggler, Low-order dPG- FEM for an elliptic PDE, Comput. Math. Appl., 68 (2014), pp. 1503-1512, doi:10.1016/j.camwa.2014.09.013.
• C. Carstensen, D. Gallistl, and M. Schedensack, Discrete reliability for Crouzeix-Raviart FEMs, SIAM J. Numer. Anal., 51 (2013), pp. 2935-2955, doi:10.1137/130915856.
• C. Carstensen and F. Hellwig, Low-order discontinuous Petrov-Galerkin finite element methods for linear elasticity, SIAM J. Numer. Anal., 54 (2016), pp. 3388-3410, doi:10.1137/15M1032582.
• C. Carstensen and F. Hellwig, Constants in discrete Poincare and Friedrichs inequalities and discrete quasi-interpolation. Submitted, 2017, arXiv:1709.00577.

This talk introduces an explicit residual-based a posteriori error analysis for the symmetric mixed finite element method in linear elasticity after Arnold-Winther with pointwise symmetric and div-conforming stress approximation. Opposed to a previous publication, the residual-based 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 mesh-refining 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 mesh-refining. A higher convergence rate for piecewise affine data is observed in the stress error and reproduced in non-smooth situations by the adaptive mesh-refining strategy.

This is joint work with Carsten Carstensen and Dietmar Gallistl.

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:

• G. Gantner, A. Haberl, D. Praetorius, B. Stiftner: Rate optimal adaptive FEM with inexact solver for nonlinear operators, accepted for publication in IMA Journal of Numerical Analysis (2017), DOI: 10.1093/imanum/drx050

We consider the adaptive finite element method for general second-order 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:

• S. Bartels, C. Carstensen, G. Dolzmann: Inhomogeneous Dirichlet conditions in a priori and a posteriori finite element error analysis, Numerische Mathematik, 99 (2004), 1-24.
• R. Sacchi, A. Veeser: Locally efficient and reliable a posteriori error estimators for Dirichlet problems, Mathematical Models and Methods in Applied Sciences (M3AS), 16 (2006), 319-346.
• M. Aurada, M. Feischl, J. Kemetmüller, M. Page, D. Praetorius: Each H^{1/2}-stable projection yields convergence and quasi-optimality of adaptive FEM with inhomogeneous Dirichlet data in Rd, Mathematical Modelling and Numerical Analysis (M2AN), 47 (2013), 1207-1235.
• M. Feischl, M. Page, D. Praetorius: Convergence and quasi-optimality of adaptive FEM with inhomogeneous Dirichlet data, Journal of Computational and Applied Mathematics, 255 (2014), 481-501.
• C. Carstensen, M. Feischl, M. Page, D. Praetorius: Axioms of adaptivity, Computers & Mathematics with Applications, 67 (2014), 1195-1253.