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Susanne C. Brenner

Publications and source records attributed to Susanne C. Brenner.

At least 19 recordsLinked to original sources
arXiv2026

A New Error Analysis for a Point Tracking Elliptic Distributed Optimal Control Problem with Pointwise Control Constraints

We develop a concise error analysis for a linear-quadratic elliptic distributed optimal control problem with point tracking and pointwise control constraints, where a standard finite element discretization and a variational discretization are treated in a unified framework. The analysis is self-contained modulo standard techniques in finite element analysis and classical results for elliptic partial differential equations.

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arXiv2025

A Spectral LOD Method for Multiscale Problems with High Contrast

We present a multiscale finite element method for a diffusion problem with rough and high contrast coefficients. The construction of the multiscale finite element space is based on the localized orthogonal decomposition methodology and it involves solutions of local finite element eigenvalue problems. We show that the performance of the multiscale finite element method is similar to the performance of standard finite element methods for the homogeneous Dirichlet boundary value problem for the Poisson equation on smooth or convex domains.} Simple explicit error estimates are established under conditions that can be verified from the outputs of the computation.

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arXiv2024

A New Error Analysis for Finite Element Methods for Elliptic Neumann Boundary Control Problems with Pointwise Control Constraints

We present a new error analysis for finite element methods for a linear-quadratic elliptic optimal control problem with Neumann boundary control and pointwise control constraints. It can be applied to standard finite element methods when the coefficient s in the elliptic operator are smooth and also to multiscale finite element methods when the coefficients are rough.

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arXiv2023

A Multiscale Finite Element Method for an Elliptic Distributed Optimal Control Problem with Rough Coefficients and Control Constraints

We construct and analyze a multiscale finite element method for an elliptic distributed optimal control problem with pointwise control constraints, where the state equation has rough coefficients. We show that the performance of the multiscale finite element method is similar to the performance of standard finite element methods for smooth problems and present corroborating numerical results.

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arXiv2020

A General Superapproximation Result

A general superapproximation result is derived in this paper which is useful for the local/interior error analysis of finite element methods.

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arXiv2020

A One Dimensional Elliptic Distributed Optimal Control Problem with Pointwise Derivative Constraints

We consider a one dimensional elliptic distributed optimal control problem with pointwise constraints on the derivative of the state. By exploiting the variational inequality satisfied by the derivative of the optimal state, we obtain higher regularity for the optimal state under appropriate assumptions on the data. We also solve the optimal control problem as a fourth order variational inequality by a $C^1$ finite element method, and present the error analysis together with numerical results.

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arXiv2020

Finite Element Methods for One Dimensional Elliptic Distributed Optimal Control Problems with Pointwise Constraints on the Derivative of the State

We investigate $C^1$ finite element methods for one dimensional elliptic distributed optimal control problems with pointwise constraints on the derivative of the state formulated as fourth order variational inequalities for the state variable. For the problem with Dirichlet boundary conditions, we use an existing $H^{\frac52-ε}$ regularity result for the optimal state to derive $O(h^{\frac12-ε})$ convergence for the approximation of the optimal state in the $H^2$ norm. For the problem with mixed Dirichlet and Neumann boundary conditions, we show that the optimal state belongs to $H^3$ under appropriate assumptions on the data and obtain $O(h)$ convergence for the approximation of the optimal state in the $H^2$ norm.

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arXiv2019

Adaptive $C^0$ interior penalty methods for Hamilton-Jacobi-Bellman equations with Cordes coefficients

In this paper we conduct a priori and a posteriori error analysis of the $C^0$ interior penalty method for Hamilton-Jacobi-Bellman equations, with coefficients that satisfy the Cordes condition. These estimates show the quasi-optimality of the method, and provide one with an adaptive finite element method. In accordance with the proven regularity theory, we only assume that the solution of the Hamilton-Jacobi-Bellman equation belongs to $H^2$.

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arXiv2019

Virtual Enriching Operators

We construct bounded linear operators that map $H^1$ conforming Lagrange finite element spaces to $H^2$ conforming virtual element spaces in two and three dimensions. These operators are useful for the analysis of nonstandard finite element methods.

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arXiv2018

Multigrid methods for saddle point problems: Karush-Kuhn-Tucker systems

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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arXiv2018

Additive Schwarz preconditioners for a state constrained elliptic distributed optimal control problem discretized by a partition of unity method

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.

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arXiv2018

A Balancing Domain Decomposition by Constraints Preconditioner for a $C^0$ Interior Penalty Method

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.

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arXiv2018

A Robust Solver for a Second Order Mixed Finite Element Method for the Cahn-Hilliard Equation

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.

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arXiv2018

Additive Schwarz Preconditioners for the Obstacle Problem of Clamped Kirchhoff Plates

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.

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