Related papers: Adaptive refinement for eigenvalue problems based …

This manuscript presents an adaptive high order discretization technique for elliptic boundary value problems. The technique is applied to an updated version of the Hierarchical Poincar\'e-Steklov (HPS) method. Roughly speaking, the HPS…

We consider adaptive finite element methods for second-order elliptic PDEs, where the arising discrete systems are not solved exactly. For contractive iterative solvers, we formulate an adaptive algorithm which monitors and steers the…

In this paper we discuss the optimal convergence of a standard adaptive scheme based on mixed finite element approximation to the solution of the eigenvalue problem associated with the Stokes equations. The proofs of the quasi-orthogonality…

Many important physical problems, such as fluid structure interaction or conjugate heat transfer, require numerical methods that compute boundary derivatives or fluxes to high accuracy. This paper proposes a novel alternative to calculating…

In this paper, we propose the unfitted spectral element method for solving elliptic interface and corresponding eigenvalue problems. The novelty of the proposed method lies in its combination of the spectral accuracy of the spectral element…

We design an adaptive finite element method to approximate the solutions of quasi-linear elliptic problems. The algorithm is based on a Ka\v{c}anov iteration and a mesh adaptation step is performed after each linear solve. The method is…

We propose an adaptive finite element algorithm to approximate solutions of elliptic problems whose forcing data is locally defined and is approximated by regularization (or mollification). We show that the energy error decay is…

In this paper, we construct an adaptive multiscale method for solving H(curl)-elliptic problems in highly heterogeneous media. Our method is based on the generalized multiscale finite element method. We will first construct a suitable…

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…

We define a generalized finite element method for the discretization of elliptic partial differential equations in heterogeneous media. An adaptive local finite element basis (AL basis) on a coarse mesh which does not resolve the matrix of…

In this work, we propose and analyze a pointwise a posteriori error estimator for simple eigenvalues of elliptic eigenvalue problems with adaptive finite element methods (AFEMs). We prove the reliability and efficiency of the residual-type…

In this article we develop a convergence theory for goal-oriented adaptive finite element algorithms designed for a class of second-order semilinear elliptic equations. We briefly discuss the target problem class, and introduce several…

Image segmentation is an inherently ill-posed problem and thus requires regularization in order to limit the search space to reasonable solutions. A majority of segmentation methods integrates these regularization terms in one way or the…

We formulate and analyze a goal-oriented adaptive finite element method for a symmetric linear elliptic partial differential equation (PDE) that can simultaneously deal with multiple linear goal functionals. In each step of the algorithm,…

In this article we develop an $hp$-adaptive refinement procedure for Trefftz discontinuous Galerkin methods applied to the homogeneous Helmholtz problem. Our approach combines not only mesh subdivision (h-refinement) and local basis…

While the exterior Helmholtz problem with Dirichlet boundary conditions is always well-posed, the associated standard boundary integral equations are not if the squared wavenumber agrees with an eigenvalue of the interior Dirichlet problem.…

We describe a strategy for solving nonlinear eigenproblems numerically. Our approach is based on the approximation of a vector-valued function, defined as solution of a non-homogeneous version of the eigenproblem. This approximation step is…

We present a perturbed subspace iteration algorithm to approximate the lowermost eigenvalue cluster of an elliptic eigenvalue problem. As a prototype, we consider the Laplace eigenvalue problem posed in a polygonal domain. The algorithm is…

The use of neural networks to approximate partial differential equations (PDEs) has gained significant attention in recent years. However, the approximation of PDEs with localised phenomena, e.g., sharp gradients and singularities, remains…

In this work, we present an adaptive unfitted finite element scheme that combines the aggregated finite element method with parallel adaptive mesh refinement. We introduce a novel scalable distributed-memory implementation of the resulting…