Related papers: High accuracy methods for eigenvalues of elliptic …

Analyses in high energy physics aim to put the Standard Model---the commonly accepted theory---to test. For convincing conclusions, analysis methods are needed which offer an unambiguous comparison between data and theory while allowing…

Low-rank tensor methods for the approximate solution of second-order elliptic partial differential equations in high dimensions have recently attracted significant attention. A critical issue is to rigorously bound the error of such…

In this paper we will develop a family of non-conforming "Crouzeix-Raviart" type finite elements in three dimensions. They consist of local polynomials of maximal degree $p\in\mathbb{N}$ on simplicial finite element meshes while certain…

We provide two new methods for computing lower bounds of eigenvalues of symmetric elliptic second-order differential operators with mixed boundary conditions of Dirichlet, Neumann, and Robin type. The methods generalize ideas of Weinstein's…

A multilevel adaptive refinement strategy for solving linear elliptic partial differential equations with random data is recalled in this work. The strategy extends the a posteriori error estimation framework introduced by Guignard and…

In this paper, we present a two-gird skill to accelerate the weak Galerkin method. By the proper use of parameters, the two-grid weak Galerkin method not only doubles the convergence rate, but also maintains the asymptotic lower bounds…

In arXiv:2307.03503 [math.NA] we commenced to study a variant of the Raviart-Thomas mixed finite element method for triangles, to solve second order elliptic equations in a curved domain with Neumann or mixed boundary conditions. It is well…

This paper presents the development and analysis of an asymptotically compatible (AC) unfitted finite element method for one-dimensional nonlocal elliptic interface problems. The proposed method achieves optimal error estimates through…

This work is devoted to the development and analysis of a linearization algorithm for microscopic elliptic equations, with scaled degenerate production, posed in a perforated medium and constrained by the homogeneous Neumann-Dirichlet…

We generalize the Brezzi-Rappaz-Raviart approximation theorem, which allows to obtain existence and a priori error estimates for approximations of solutions to some nonlinear partial differential equations. Our contribution lies in the fact…

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…

We extend the standard notion of self-concordance to non-convex optimization and develop a family of second-order algorithms with global convergence guarantees. In particular, two function classes -- \textit{weakly self-concordant}…

This paper generalizes the non-conforming FEM of Crouzeix and Raviart and its fundamental projection property by a novel mixed formulation for the Poisson problem based on the Helmholtz decomposition. The new formulation allows for ansatz…

We propose and analyze a finite element method for the Oseen eigenvalue problem. This problem is an extension of the Stokes eigenvalue problem, where the presence of the convective term leads to a non-symmetric problem and hence, to complex…

We consider mixed finite element approximation of a singularly perturbed fourth-order elliptic problem with two different boundary conditions, and present a new measure of the error, whose components are balanced with respect to the…

We verify quasi-optimality of the Crouzeix-Raviart FEM for nonlinear problems of $p$-Laplace type. More precisely, we show that the error of the Crouzeix-Raviart FEM with respect to a quasi-norm is bounded from above by a uniformly bounded…

In this work we develop and analyze an adaptive finite element method for efficiently solving electrical impedance tomography -- a severely ill-posed nonlinear inverse problem for recovering the conductivity from boundary voltage…

Most classical finite element schemes for the (Navier-)Stokes equations are neither pressure-robust, nor are they inf-sup stable on general anisotropic triangulations. A lack of pressure-robustness may lead to large velocity errors,…

We develop an essentially optimal finite element approach for solving ergodic stochastic two-scale elliptic equations whose two-scale coefficient may depend also on the slow variable. We solve the limiting stochastic two-scale homogenized…

In this paper, we develop a regularized higher-order Taylor based method for solving composite (e.g., nonlinear least-squares) problems. At each iteration, we replace each smooth component of the objective function by a higher-order Taylor…