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It is shown how mixed finite element methods for symmetric positive definite eigenvalue problems related to partial differential operators can provide guaranteed lower eigenvalue bounds. The method is based on a classical compatibility…
We consider the finite element solution of the vector Laplace equation on a domain in two dimensions. For various choices of boundary conditions, it is known that a mixed finite element method, in which the rotation of the solution is…
Discrete de Rham (DDR) methods provide non-conforming but compatible approximations of the continuous de Rham complex on general polytopal meshes. Owing to the non-conformity, several challenges arise in the analysis of these methods. In…
We propose a linear finite-element discretization of Dirichlet problems for static Hamilton-Jacobi equations on unstructured triangulations. The discretization is based on simplified localized Dirichlet problems that are solved by a local…
For a generalized Hodge Laplace equation, we prove the quasi-optimal convergence rate of an adaptive mixed finite element method. This adaptive method can control the error in the natural mixed variational norm when the space of harmonic…
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 derive optimal and asymptotically exact a posteriori error estimates for the approximation of the Laplace eigenvalue problem. To do so, we combine two results from the literature. First, we use the hypercircle techniques developed for…
We solve elliptic systems of equations posed on highly heterogeneous materials. Examples of this class of problems are composite structures and geological processes. We focus on a model problem which is a second-order elliptic equation with…
The state-of-the art proof of a global inf-sup condition on mixed finite element schemes does not allow for an analysis of truly indefinite, second-order linear elliptic PDEs. This paper, therefore, first analyses a nonconforming finite…
We develop in this work the first polytopal complexes of differential forms. These complexes, inspired by the Discrete De Rham and the Virtual Element approaches, are discrete versions of the de Rham complex of differential forms built on…
Robust mixed finite element methods are developed for a quad-curl singular perturbation problem. Lower order H(grad curl)-nonconforming but H(curl)-conforming finite elements are constructed, which are extended to nonconforming finite…
This paper constructs a unified family of nonconforming finite element spaces for $H\Lambda^k$ in $\mathbb{R}^n$ ($0\leqslant k\leqslant n$, $n\geqslant 1$). The spaces employ piecewise Whitney forms as shape functions, and include the…
We consider linear and obstacle problems driven by a nonlocal integral operator, for which nonlocal interactions are restricted to a ball of finite radius. These type of operators are used to model anomalous diffusion and, for a special…
Based on the work of Xu and Zhou [Math.Comput., 69(2000), pp.881-909], we establish new three-level and multilevel finite element discretizations by local defect-correction technique. Theoretical analysis and numerical experiments show that…
We introduce an adaptive superconvergent finite element method for a class of mixed formulations to solve partial differential equations involving a diffusion term. It combines a superconvergent postprocessing technique for the primal…
We consider an elliptic partial differential equation in non-divergence form with a random diffusion matrix and random forcing term. To address this, we propose a mixed-type continuous finite element discretization in the physical domain,…
In this paper, we study a generalized finite element method for solving second-order elliptic partial differential equations with rough coefficients. The method uses local approximation spaces computed by solving eigenvalue problems on…
In this paper, we study local convergence of high-order Tensor Methods for solving convex optimization problems with composite objective. We justify local superlinear convergence under the assumption of uniform convexity of the smooth…
It is shown that the h-adaptive mixed finite element method for the discretization of eigenvalue clusters of the Laplace operator produces optimal convergence rates in terms of nonlinear approximation classes. The results are valid for the…
In this article, we consider a combination of local and nonlocal Laplace equation with singular nonlinearities. For such mixed problems, we establish existence of at least one weak solution for a parameter dependent singular nonlinearity…