Showing papers by "Susanne C. Brenner published in 2011"
TL;DR: In this paper, the authors developed and analyzed C(0) penalty methods for the fully nonlinear Monge-Ampere equation det(D(2)u) = f in two dimensions, where the key idea is to build discretizations such that the resulting discrete linearizations are symmetric, stable, and consistent with the continuous linearization.
Abstract: In this paper, we develop and analyze C(0) penalty methods for the fully nonlinear Monge-Ampere equation det(D(2)u) = f in two dimensions. The key idea in designing our methods is to build discretizations such that the resulting discrete linearizations are symmetric, stable, and consistent with the continuous linearization. We are then able to show the well-posedness of the penalty method as well as quasi-optimal error estimates using the Banach fixed-point theorem as our main tool. Numerical experiments are presented which support the theoretical results.
102 citations
01 Jan 2011
TL;DR: C 0 interior penalty methods are discontinuous Galerkin methods for fourth order problems including a priori error analysis, a posteriorierror analysis and fast solution techniques.
Abstract: C 0 interior penalty methods are discontinuous Galerkin methods for fourth order problems. In this article we discuss various aspects of such methods including a priori error analysis, a posteriori error analysis and fast solution techniques.
52 citations
TL;DR: A posteriori error analysis techniques are used to show that the method converges in the energy norm uniformly with respect to the perturbation parameter under minimal regularity assumptions, and the convergence of the numerical solution to the unperturbed second order problem.
Abstract: In this paper, we develop a $\mathcal{C}^0$ interior penalty method for a fourth order singular perturbation elliptic problem in two dimensions on polygonal domains. Using some a posteriori error analysis techniques, we are able to show that the method converges in the energy norm uniformly with respect to the perturbation parameter under minimal regularity assumptions. In addition, we analyze the convergence of the numerical solution to the unperturbed second order problem. Finally, we perform some numerical experiments that back up the theoretical results.
47 citations
TL;DR: This work proposes and analyzes several two-level additive Schwarz preconditioners for a weakly over-penalized symmetric interior penalty method for second order elliptic boundary value problems.
Abstract: We propose and analyze several two-level additive Schwarz preconditioners for a weakly over-penalized symmetric interior penalty method for second order elliptic boundary value problems. We also report numerical results that illustrate the parallel performance of these preconditioners.
45 citations
TL;DR: Optimal order error estimates are derived in both the energy norm and the L2 norm, and the uniform convergence of V-cycle, F-cycle and W-cycle multigrid algorithms for the resulting discrete problems are established.
Abstract: We study a class of symmetric discontinuous Galerkin methods on graded meshes Optimal order error estimates are derived in both the energy norm and the L 2 norm, and we establish the uniform convergence of V-cycle, F-cycle and W-cycle multigrid algorithms for the resulting discrete problems Numerical results that confirm the theoretical results are also presented
32 citations
TL;DR: A new numerical approach for two-dimensional Maxwell’s equations that is based on the Hodge decomposition for divergencefree vector fields is proposed, which can be obtained by solving standard second order scalar elliptic boundary value problems.
Abstract: We propose a new numerical approach for two-dimensional Maxwell’s equations that is based on the Hodge decomposition for divergencefree vector fields. In this approach an approximate solution for Maxwell’s equations can be obtained by solving standard second order scalar elliptic boundary value problems. This new approach is illustrated by a P1 finite element method.
25 citations
TL;DR: It is proved that the condition number of the preconditioned system is bounded by C[1 + (H/δ)], where H represents the coarse mesh size,δ measures the overlap among the subdomains, and the constant C is independent of H, δ, the fine mesh size h and the number of subdomain Ns .
Abstract: Abstract We propose and analyze overlapping two-level additive Schwarz preconditioners for the local discontinuous Galerkin discretization. We prove that the condition number of the preconditioned system is bounded by C[1 + (H/δ)], where H represents the coarse mesh size, δ measures the overlap among the subdomains, and the constant C is independent of H, δ, the fine mesh size h and the number of subdomains Ns . Numerical results are presented showing the scalability of the method.
13 citations
01 Nov 2011
TL;DR: A nonoverlapping domain decomposition preconditioner for the weakly over-penalized symmetric interior penalty (WOPSIP) method andoretical results on the condition number estimate of the preconditionsed system will be presented along with numerical results.
Abstract: In this talk we will discuss a nonoverlapping domain decomposition preconditioner for the weakly over-penalized symmetric interior penalty (WOPSIP) method. The WOPSIP method belongs to the family of discontinuous finite element methods. The preconditioner for theWOPSIP method is based on the balancing domain decomposition by constraints methodology. Theoretical results on the condition number estimate of the preconditioned system will be presented along with numerical results.
4 citations