Showing papers by "Susanne C. Brenner published in 2018"
TL;DR: A model Poisson problem in [Formula: see text] is considered and error estimates for virtual element methods on polygonal or polyhedral meshes that can contain small edges or small faces are established.
Abstract: We consider a model Poisson problem in ℝd (d = 2, 3) and establish error estimates for virtual element methods on polygonal or polyhedral meshes that can contain small edges (d = 2) or small faces ...
152 citations
TL;DR: This work designs and analyzes interior penalty methods for an elliptic distributed optimal control problem on nonconvex polygonal domains with pointwise state constraints.
Abstract: We design and analyze $C^0$ interior penalty methods for an elliptic distributed optimal control problem on nonconvex polygonal domains with pointwise state constraints.
19 citations
TL;DR: In this paper, a Morley finite element method for an elliptic distributed optimal control problem with pointwise state and control constraints on convex polygonal domains is proposed, which is based on the formulation of the optimization problem as a fourth order variational inequality.
Abstract: We design and analyze a Morley finite element method for an elliptic distributed optimal control problem with pointwise state and control constraints on convex polygonal domains. It is based on the formulation of the optimal control problem as a fourth order variational inequality. Numerical results that illustrate the performance of the method are also presented.
14 citations
TL;DR: In this paper, a robust solver for a mixed finite element convex splitting scheme for the Cahn-Hilliard equation is presented, whose performance is independent of the spacial mesh size and the time step size for a given interfacial width parameter.
Abstract: We develop a robust solver for a mixed finite element convex splitting scheme for the Cahn–Hilliard equation. The key ingredient of the solver is a preconditioned minimal residual algorithm (with a multigrid preconditioner) whose performance is independent of the spacial mesh size and the time step size for a given interfacial width parameter. The dependence on the interfacial width parameter is also mild.
13 citations
Book•
01 Jan 2018
TL;DR: These are the proceedings of the 24th International Conference on Domain Decomposition Methods in Science and Engineering, which was held in Svalbard, Norway in February 2017.
Abstract: These are the proceedings of the 24th International Conference on Domain Decomposition Methods in Science and Engineering, which was held in Svalbard, Norway in February 2017. Domain decomposition methods are iterative methods for solving the often very large systems of equations that arise when engineering problems are discretized, frequently using finite elements or other modern techniques. These methods are specifically designed to make effective use of massively parallel, high-performance computing systems. The book presents both theoretical and computational advances in this domain, reflecting the state of art in 2017.
8 citations
TL;DR: A nonconforming finite element approximation of the vibration modes of an acoustic fluid-structure interaction based on an irrotational fluid displacement formulation and hence it is free of spurious eigenmodes is studied.
Abstract: Abstract We study a nonconforming finite element approximation of the vibration modes of an acoustic fluid-structure interaction. Displacement variables are used for both the fluid and the solid. The numerical scheme is based on an irrotational fluid displacement formulation and hence it is free of spurious eigenmodes. The method uses weakly continuous P1{P_{1}} vector fields for the fluid and classical piecewise linear elements for the solid, and it has O(h2){O(h^{2})} convergence for the eigenvalues on properly graded meshes. The theoretical results are confirmed by numerical experiments.
8 citations
TL;DR: In this article, the authors design and analyze multigrid methods for saddle point problems resulting from Raviart-Thomas-Nedelec mixed finite element methods for the Darcy system in porous media flow.
Abstract: We design and analyze multigrid methods for the saddle point problems resulting from Raviart–Thomas–Nedelec mixed finite element methods for the Darcy system in porous media flow. Uniform convergence of the W-cycle algorithm in a nonstandard energy norm is established. Extensions to general second order elliptic problems are also addressed.
5 citations
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TL;DR: Additive Schwarz preconditioners are presented for a class of elliptic optimal control problems discretized by a partition of unity method and condition number estimates are given.
Abstract: We present additive Schwarz preconditioners for a class of elliptic optimal control problems discretized by a partition of unity method. The discrete problem is solved by a primal-dual active set algorithm, where the auxiliary system in each iteration is solved by a preconditioned conjugate gradient method based on additive Schwarz preconditioners. Condition number estimates are given and verified by a numerical example.
4 citations
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TL;DR: In this article, additive Schwarz preconditioners for the systems that appear in each iteration of the primal-dual active set algorithm are developed and analyzed for the system that appears in the first iteration.
Abstract: When the obstacle problem of clamped Kirchhoff plates is discretized by a partition of unity method, the resulting discrete variational inequalities can be solved by a primal-dual active set algorithm. In this paper we develop and analyze additive Schwarz preconditioners for the systems that appear in each iteration of the primal-dual active set algorithm. Numerical results that corroborate the theoretical estimates are also presented.
4 citations
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TL;DR: In this paper, a robust solver for a second order mixed finite element splitting scheme for the Cahn-Hilliard equation is presented, whose performance is independent of the spacial mesh size and the time step size for a given interfacial width parameter.
Abstract: We develop a robust solver for a second order mixed finite element splitting scheme for the Cahn-Hilliard equation. This work is an extension of our previous work in which we developed a robust solver for a first order mixed finite element splitting scheme for the Cahn-Hilliard equaion. The key ingredient of the solver is a preconditioned minimal residual algorithm (with a multigrid preconditioner) whose performance is independent of the spacial mesh size and the time step size for a given interfacial width parameter. The dependence on the interfacial width parameter is also mild.
4 citations
TL;DR: Uniform convergence of the V‐cycle methods on bounded convex hexahedral domains (rectangular boxes) is proved and it is proved that the smoothers in the multigrid methods involve nonoverlapping domain decomposition preconditioners that are based on substructuring.
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TL;DR: The uniform convergence of the $W$-cycle algorithm is proved and the performance of the V and W-cycle algorithms in two and three dimensions is demonstrated through numerical experiments.
Abstract: We construct multigrid methods for an elliptic distributed optimal control problem that are robust with respect to a regularization parameter. We prove the uniform convergence of the $W$-cycle algorithm and demonstrate the performance of $V$-cycle and $W$-cycle algorithms in two and three dimensions through numerical experiments.
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TL;DR: It is proved that the condition number of the preconditioned system is bounded by $C (1+\ln (H/h))^2", where h is the mesh size of the triangulation, H is the typical diameter of subdomains, and the positive constant $C$ is independent of h and $H$.
Abstract: We develop a nonoverlapping domain decomposition preconditioner for the $C^0$ interior penalty method, a discontinuous Galerkin method, for the biharmonic problem. The preconditioner is based on balancing domain decomposition by constraints (BDDC). We prove that the condition number of the preconditioned system is bounded by $C (1+\ln (H/h))^2$, where $h$ is the mesh size of the triangulation, $H$ is the typical diameter of subdomains, and the positive constant $C$ is independent of $h$ and $H$. Numerical experiments are also represented to corroborate the theoretical result.