Rate-optimal goal-oriented adaptive FEM for semilinear elliptic PDEs

On goal-oriented adaptivity for elliptic optimal control problems

The paper proposes goal-oriented error estimation and mesh refinement for optimal control problems with elliptic PDE constraints using the value of the reduced cost functional as quantity of interest. Error representation, hierarchical error estimators, and greedy-style error indicators are derived and compared to their counterparts when using the all-at-once cost functional as quantity of inte...

Newton's Method and Morse Index for semilinear Elliptic PDEs

In this paper we primarily consider the family of elliptic PDEs ∆u+ f(u) = 0 on the square region Ω = (0, 1)× (0, 1) with zero Dirichlet boundary condition. Following our previous analysis and numerical approximations which relied on the variational characterization of solutions as critical points of an “action” functional, we consider Newton’s method on the gradient of that functional. We use ...

A heterogeneous stochastic FEM framework for elliptic PDEs

We introduce a new concept of sparsity for the stochastic elliptic operator −div (a(x,ω)∇(·)), which reflects the compactness of its inverse operator in the stochastic direction and allows for spatially heterogeneous stochastic structure. This new concept of sparsity motivates a heterogeneous stochastic finite element method (HSFEM) framework for linear elliptic equations, which discretizes the...

Convergence of goal-oriented adaptive finite element methods for semilinear problems

In this article we develop convergence theory for a class of goal-oriented adaptive finite element algorithms for second order semilinear elliptic equations. We first introduce several approximate dual problems, and briefly discuss the target problem class. We then review some standard facts concerning conforming finite element discretization and error-estimate-driven adaptive finite element me...

Optimal Control of Semilinear Elliptic Obstacle Problems