Related papers: A Nitsche-eXtended finite element method for distr…
We adopt the integral definition of the fractional Laplace operator and analyze an optimal control problem for a fractional semilinear elliptic partial differential equation (PDE); control constraints are also considered. We establish the…
We consider the finite element discretization and the iterative solution of singularly perturbed elliptic reaction-diffusion equations in three-dimensional computational domains. These equations arise from the optimality conditions for…
We consider the reliable implementation of high-order unfitted finite element methods on Cartesian meshes with hanging nodes for elliptic interface problems. We construct a reliable algorithm to merge small interface elements with their…
Optimization problems with $L^1$-control cost functional subject to an elliptic partial differential equation (PDE) are considered. However, different from the finite dimensional $l^1$-regularization optimization, the resulting discretized…
This work proposes a nonlinear finite element method whose nodal values preserve bounds known for the exact solution. The discrete problem involves a nonlinear projection operator mapping arbitrary nodal values into bound-preserving ones…
We present a novel approach that integrates unfitted finite element methods and neural networks to approximate partial differential equations on complex geometries. Easy-to-generate background meshes (e.g., a simple Cartesian mesh) that cut…
We consider an optimal control problem that entails the minimization of a nondifferentiable cost functional, fractional diffusion as state equation and constraints on the control variable. We provide existence, uniqueness and regularity…
We study a linear-quadratic optimal control problem involving a parabolic equation with fractional diffusion and Caputo fractional time derivative of orders $s \in (0,1)$ and $\gamma \in (0,1]$, respectively. The spatial fractional…
We present a new composite mesh finite element method for fluid--structure interaction problems. The method is based on surrounding the structure by a boundary-fitted fluid mesh which is embedded into a fixed background fluid mesh. The…
This work establishes a general stochastic maximum principle for partially observed optimal control of semi-linear stochastic partial differential equations in a nonconvex control domain. The state evolves in a Hilbert space driven by a…
In this paper, we propose a novel mesh-free numerical method for solving the elliptic interface problems based on deep learning. We approximate the solution by the neural networks and, since the solution may change dramatically across the…
Nitsche's method is a popular approach to implement Dirichlet-type boundary conditions in situations where a strong imposition is either inconvenient or simply not feasible. The method is widely applied in the context of unfitted finite…
This work is motivated by the need to study the impact of data uncertainties and material imperfections on the solution to optimal control problems constrained by partial differential equations. We consider a pathwise optimal control…
We study optimal control problems that are governed by semilinear elliptic partial differential equations that involve non-Lipschitzian nonlinearities. It is shown that, for a certain class of such PDEs, the solution map is Fr\'{e}chet…
We present guidelines for deriving new Nitsche Finite Element Methods to enforce equality and inequality constraints that act on the value of the unknown mechanical quantity. We first formulate the problem as a stabilized finite element…
We analyze space-time finite element methods for the numerical solution of distributed parabolic optimal control problems with energy regularization in the Bochner space $L^2(0,T;H^{-1}(\Omega))$. By duality, the related norm can be…
This work presents and analyzes space-time finite element methods on fully unstructured simplicial space-time meshes for the numerical solution of parabolic optimal control problems. Using Babu\v{s}ka's theorem, we show well-posedness of…
In this paper we analyze the Schwarz alternating method for unconstrained elliptic optimal control problems. We discuss the convergence properties of the method in the continuous case first and then apply the arguments to the finite…
In this paper error analysis for finite element discretizations of Dirichlet boundary control problems is developed. For the first time, optimal discretization error estimates are established in the case of three dimensional polyhedral and…
We consider the reliable implementation of an adaptive high-order unfitted finite element method on Cartesian meshes for solving elliptic interface problems with geometrically curved singularities. We extend our previous work on the…