Showing papers by "Susanne C. Brenner published in 2014"

A Quadratic C0 Interior Penalty Method for an Elliptic Optimal Control Problem with State Constraints

Abstract: We consider an elliptic distributed optimal control problem on convex polygonal domains with pointwise state constraints and solve it as a fourth order variational inequality for the state by a quadratic C 0 interior penalty method. The error for the state in an H 2-like energy norm is O(h α ) on quasi-uniform meshes (where α ∈ (0, 1] is determined by the interior angles of the domain) and O(h) on graded meshes. The error for the control in the L 2 norm has the same behavior. Numerical results that illustrate the performance of the method are also presented.

Multigrid methods for saddle point problems: Stokes and Lamé systems

Abstract: We develop new multigrid methods for a class of saddle point problems that include the Stokes system in fluid flow and the Lame system in linear elasticity as special cases. The new smoothers in the multigrid methods involve optimal preconditioners for the discrete Laplace operator. We prove uniform convergence of the $$W$$ W -cycle algorithm in the energy norm and present numerical results for $$W$$ W -cycle and $$V$$ V -cycle algorithms.

A Mixed Finite Element Method for the Stokes Equations Based on a Weakly Over-Penalized Symmetric Interior Penalty Approach

Abstract: We present a mixed finite element method for the steady-state Stokes equations where the discrete bilinear form for the velocity is obtained by a weakly over-penalized symmetric interior penalty approach. We show that this mixed finite element method is inf-sup stable and has optimal convergence rates in both the energy norm and the $$L_2$$ L 2 norm on meshes that can contain hanging nodes. We present numerical experiments illustrating these results, explore a very simple adaptive algorithm that uses meshes with hanging nodes, and introduce a simple but scalable parallel solver for the method.

A One-Level Additive Schwarz Preconditioner for a Discontinuous Petrov–Galerkin Method

Abstract: Discontinuous Petrov–Galerkin (DPG) methods are new discontinuous Galerkin methods [3–8] with interesting properties. In this article we consider a domain decomposition preconditioner for a DPG method for the Poisson problem.

Piecewise ¹ functions and vector fields associated with meshes generated by independent refinements

Abstract: Abstract. We consider piecewise H1 functions and vector fields associated with a class of meshes generated by independent refinements and show that they can be effectively analyzed in terms of the number of refinement levels and the shape regularity of the subdomains that appear in the meshes. We derive Poincaré-Friedrichs inequalities and Korn’s inequalities for such meshes and discuss an application to a discontinuous finite element method.