Related papers: Finite element error estimates for the nonlinear S…
We present a novel and comparative analysis of finite element discretizations for a nonlinear Rosenau-Burgers model including a biharmonic term. We analyze both continuous and mixed finite element approaches, providing stability, existence,…
The paper studies the well-posedness and optimal error estimates of spectral finite element approximations for the boundary value problems of semi-linear elliptic SPDEs driven by white or colored Gaussian noises. The noise term is…
Optimal convergence rates of adaptive finite element methods are well understood in terms of the axioms of adaptivity. One key ingredient is the discrete reliability of a residual-based a posteriori error estimator, which controls the error…
This work considers the nodal finite element approximation of peridynamics, in which the nodal displacements satisfy the peridynamics equation at each mesh node. For the nonlinear bond-based peridynamics model, it is shown that, under the…
This work develops an epsilon-uniform finite element method for singularly perturbed boundary value problems. A surprising and remarkable observation is illustrated: By moving one node arbitrarily in between its adjacent nodes, the new…
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…
The Morley finite element method (FEM) is attractive for semilinear problems with the biharmonic operator as a leading term in the stream function vorticity formulation of 2D Navier-Stokes problem and in the von K\'{a}rm\'{a}n equations.…
In this article a special class of nonlinear optimal control problems involving a bilinear term in the boundary condition is studied. These kind of problems arise for instance in the identification of an unknown space-dependent Robin…
This paper considers the finite element approximation to parabolic optimal control problems with measure data in a nonconvex polygonal domain. Such problems usually possess low regularity in the state variable due to the presence of measure…
For elliptic interface problems in two- and three-dimensions, this paper establishes a priori error estimates for Crouzeix-Raviart nonconforming, Raviart-Thomas mixed, and discontinuous Galerkin finite element approximations. These…
This paper derives an a posteriori error estimator for the nonlinear first-order optimality conditions associated with the electrically and flexoelectrically coupled Frank-Oseen model of liquid crystals, building on previous results for…
We study the asymptotic error between the finite element solutions of nonlocal models with a bounded interaction neighborhood and the exact solution of the limiting local model. The limit corresponds to the case when the horizon parameter,…
A numerical method for approximating weak solutions of an aggregation equation with degenerate diffusion is introduced. The numerical method consists of a stabilized finite element method together with a mass lumping technique and an extra…
This paper deals with the backward Euler method applied to semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive noise. The SPDE is discretized in space by the finite element method and in time by the…
The recently proposed statistical finite element (statFEM) approach synthesises measurement data with finite element models and allows for making predictions about the unknown true system response. We provide a probabilistic error analysis…
In this paper, we will consider an $hp$-finite elements discretization of a highly indefinite Helmholtz problem by some dG formulation which is based on the ultra-weak variational formulation by Cessenat and Depr\'{e}s. We will introduce an…
In the spirit of the classical work of P. H. Rabinowitz on nonlinear Schr\"odinger equations, we prove existence of mountain-pass solutions and least energy solutions to the nonlinear Schr\"odinger-Poisson system \begin{equation}\nonumber…
This paper introduces a discretization-accurate stopping criterion of symmetric iterative methods for solving systems of algebraic equations resulting from the finite element approximation. The stopping criterion consists of the evaluations…
We propose a novel a posteriori error estimator for conforming finite element discretizations of two- and three-dimensional Helmholtz problems. The estimator is based on an equilibrated flux that is computed by solving patchwise mixed…
In this paper, we present an implicit Crank-Nicolson finite element (FE) scheme for solving a nonlinear Schr\"odinger-type system, which includes Schr\"odinger-Helmholz system and Schr\"odinger-Poisson system. In our numerical scheme, we…