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The main aim of this article is to analyze mixed finite element method for the second order Dirichlet boundary control problem. Therein, we develop both a priori and a posteriori error analysis using the energy space based approach. We…
A general adaptive refinement strategy for solving linear elliptic partial differential equation with random data is proposed and analysed herein. The adaptive strategy extends the a posteriori error estimation framework introduced by…
The purpose of this work is the design and analysis of a reliable and efficient a posteriori error estimator for the so-called pointwise tracking optimal control problem. This linear-quadratic optimal control problem entails the…
In this work, we consider an optimal control problem subject to a nonlinear PDE constraint and apply it to the regularized $p$-Laplace equation. To this end, a reduced unconstrained optimization problem in terms of the control variable is…
Physics-informed deep learning has emerged as a promising alternative for solving partial differential equations. However, for complex problems, training these networks can still be challenging, often resulting in unsatisfactory accuracy…
We consider initial value problems where we are interested in a quantity of interest (QoI) that is the integral in time of a functional of the solution of the IVP. For these, we look into local error based time adaptivity. We derive a goal…
The paper deals with the a posteriori error analysis of a virtual element method for the Steklov eigenvalue problem. The virtual element method has the advantage of using general polygonal meshes, which allows implementing very efficiently…
A nonlinear Helmholtz (NLH) equation with high frequencies and corner singularities is discretized by the linear finite element method (FEM). After deriving some wave-number-explicit stability estimates and the singularity decomposition for…
This work deals with the design optimization of electrical machines under the consideration of manufacturing uncertainties. In order to efficiently quantify the uncertainty, blackbox machine learning methods are employed. A multi-objective…
We describe and analyze a hybrid finite element/neural network method for predicting solutions of partial differential equations. The methodology is designed for obtaining fine scale fluctuations from neural networks in a local manner. The…
Adaptive multilevel finite element methods are developed and analyzed for certain elliptic systems arising in geometric analysis and general relativity. This class of nonlinear elliptic systems of tensor equations on manifolds is first…
In this paper, a parameter-uniform fitted mesh finite difference scheme is constructed and analyzed for a class of singularly perturbed interior turning point problems. The solution of this class of turning point problem possess two outflow…
Port-Hamiltonian systems provide a highly-structured framework for modeling of physical systems. By definition, they encode a balance equation relating energy changes to supplied and dissipated energy. Capturing this energy balance in…
We propose and analyze novel adaptive algorithms for the numerical solution of elliptic partial differential equations with parametric uncertainty. Four different marking strategies are employed for refinement of stochastic Galerkin finite…
A new anisotropic mesh adaptation strategy for finite element solution of elliptic differential equations is presented. It generates anisotropic adaptive meshes as quasi-uniform ones in some metric space, with the metric tensor being…
We analyze adaptive mesh-refining algorithms in the frame of boundary element methods (BEM) and the coupling of finite elements and boundary elements (FEM-BEM). Adaptivity is driven by the two-level error estimator proposed by Ernst P.…
We devise an a posteriori error estimator for an affine optimal control problem subject to a semilinear elliptic PDE and control constraints. To approximate the problem, we consider a semidiscrete scheme based on the variational…
In this paper, we design the first residual type a posteriori error estimator for mixed interior penalty discontinuous Galerkin method for the H(curl)-elliptic problems. Then we prove that our residual based a posteriori error indicator is…
The quantum many-body problem can be rephrased as a variational determination of the two-body reduced density matrix, subject to a set of N-representability constraints. The mathematical problem has the form of a semidefinite program. We…
We consider the Galerkin boundary element method (BEM) for weakly-singular integral equations of the first-kind in 2D. We analyze some residual-type a posteriori error estimator which provides a lower as well as an upper bound for the…