Browsing by Author "Karkulik, Michael"
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- ItemA ROBUST DPG METHOD FOR SINGULARLY PERTURBED REACTION-DIFFUSION PROBLEMS(2017) Heuer, Norbert; Karkulik, MichaelWe present and analyze a discontinuous Petrov-Galerkin method with optimal test functions for a reaction-dominated diffusion problem in two and three space dimensions. We start with an ultraweak formulation that comprises parameters alpha, beta to allow for general epsilon-dependent weightings of three field variables (epsilon being the small diffusion parameter). Specific values of alpha and beta imply robustness of the method, that is, a quasi-optimal error estimate with a constant that is independent of epsilon. Moreover, these values lead to a norm for the field variables that is known to be balanced in epsilon for model problems with typical boundary layers. Several numerical examples underline our theoretical estimates and reveal stability of approximations even for very small epsilon.
- ItemAdaptive Boundary Element Methods A Posteriori Error Estimators, Adaptivity, Convergence, and Implementation(2015) Feischl, M.; Führer, Thomas; Heuer, Norbert; Karkulik, Michael; Praetorius, D.
- ItemAdaptive Crouzeix-Raviart boundary element method(2015) Heuer, Norbert; Karkulik, Michael
- ItemAnalysis of Backward Euler Primal DPG Methods(2021) Führer, Thomas; Heuer, Norbert; Karkulik, MichaelWe analyze backward Euler time stepping schemes for a primal DPG formulation of a class of parabolic problems. Optimal error estimates are shown in a natural norm and in the L-2 norm of the field variable. For the heat equation the solution of our primal DPG formulation equals the solution of a standard Galerkin scheme and, thus, optimal error bounds are found in the literature. In the presence of advection and reaction terms, however, the latter identity is not valid anymore and the analysis of optimal error bounds requires to resort to elliptic projection operators. It is essential that these operators be projections with respect to the spatial part of the PDE, as in standard Galerkin schemes, and not with respect to the full PDE at a time step, as done previously.
- ItemCombining the DPG Method with Finite Elements(2018) Fuehrer, Thomas; Heuer, Norbert; Karkulik, Michael; Rodriguez, Rodolfo
- ItemConvergence of Adaptive 3D BEM for Weakly Singular Integral Equations Based on Isotropic Mesh-Refinement(2013) Karkulik, Michael; Of, Günther; Praetorius, Dirk
- ItemDiscontinuous Petrov-Galerkin boundary elements(2017) Heuer, Norbert; Karkulik, Michael
- ItemDPG Method with Optimal Test Functions for a Fractional Advection Diffusion Equation(2017) Ervin, V.; Führer, Thomas; Heuer, Norbert; Karkulik, Michael
- ItemDPG method with optimal test functions for a transmission problem(2015) Heuer, Norbert; Karkulik, Michael
- ItemEfficiency and optimality of some weighted-residual error estimator for adaptive 2D boundary element methods(2013) Aurada, M.; Feischl, M.; Führer, T.; Karkulik, Michael; Praetorius, D.We prove convergence and quasi-optimality of a lowest-order adaptive boundary element method for a weakly-singular integral equation in 2D. The adaptive mesh-refinement is driven by the weighted-residual error estimator. By proving that this estimator is not only reliable, but under some regularity assumptions on the given data also efficient on locally refined meshes, we characterize the approximation class in terms of the Galerkin error only. In particular, this yields that no adaptive strategy can do better, and the weighted-residual error estimator is thus an optimal choice to steer the adaptive mesh-refinement. As a side result, we prove a weak form of the saturation assumption.
- ItemEnergy norm based error estimators for adaptive BEM for hypersingular integral equations(2015) Aurada, Markus; Feischl, Michael; Füehrer, Thomas; Karkulik, Michael; Praetorius, Dirk
- ItemHILBERT - a MATLAB implementation of adaptive 2D-BEM(2014) Aurada, Markus; Ebner, Michael; Feischl, Michael; Ferraz-Leite, Samuel; Führer, Thomas; Goldenits, Petra; Karkulik, Michael; Mayr, Markus; Praetorius, Dirk
- ItemLocal high-order regularization and applications to hp-methods(2015) Karkulik, Michael; Melenk, J.
- ItemLOCAL INVERSE ESTIMATES FOR NON-LOCAL BOUNDARY INTEGRAL OPERATORS(2017) Aurada, M.; Feischl, M.; Karkulik, Michael; Melenk, J.; Praetorius, D.; Führer, Thomas
- ItemMINRES for Second-Order PDEs with Singular Data(2022) Führer Thomas; Heuer, Norbert; Karkulik, MichaelMinimum residual methods such as the least-squares finite element method (FEM) or the discontinuous Petrov-Galerkin (DPG) method with optimal test functions usually exclude singular data, e.g., non-square-integrable loads. We consider a DPG method and a least-squares FEM for the Poisson problem. For both methods we analyze regularization approaches that allow the use of H-1 loads and also study the case of point loads. For all cases we prove appropriate convergence orders. We present various numerical experiments that confirm our theoretical results. Our approach extends to general well-posed second-order problems.
- ItemNew a priori analysis of first-order system least-squares finite element methods for parabolic problems(2019) Führer, Thomas; Karkulik, Michael
- ItemNote on discontinuous trace approximation in the practical DPG method(2014) Heuer, Norbert; Karkulik, Michael; Sayas, F.
- ItemOn 2D Newest Vertex Bisection : Optimality of Mesh-Closure and H (1)-Stability of L (2)-Projection(2013) Karkulik, Michael; Pavlicek, D.; Praetorius, D.
- ItemON THE COUPLING OF DPG AND BEM(2017) Führer, Thomas; Heuer, Norbert; Karkulik, Michael
- ItemQUASI-OPTIMAL CONVERGENCE RATE FOR AN ADAPTIVE BOUNDARY ELEMENT METHOD(2013) Feischl, M.; Karkulik, Michael; Melenk, J.; Praetorius, D.