Related papers: Adaptive finite element approximation for steady-s…
A widely used electrostatics model in the biomolecular modeling community, the nonlinear Poisson-Boltzmann equation, along with its finite element approximation, are analyzed in this paper. A regularized Poisson-Boltzmann equation is…
Under compressive creep, visco-plastic solids experiencing internal mass transfer processes have been recently proposed to accommodate singular cnoidal wave solutions, as material instabilities at the stationary wave limit. These…
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 a posteriori error estimates are studied for a class of nonlinear stead-state Poisson-Nernst-Planck equations, which are a coupled system consisting of the Nernst-Planck equation and the Poisson equation. Both the global upper bounds…
In this article we study adaptive finite element methods (AFEM) with inexact solvers for a class of semilinear elliptic interface problems. We are particularly interested in nonlinear problems with discontinuous diffusion coefficients, such…
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…
In this paper, we study adaptive finite element approximations in a perturbation framework, which makes use of the existing adaptive finite element analysis of a linear symmetric elliptic problem. We prove the convergence and complexity of…
This paper details the development and application of an $h$-adaptive finite element method for the numerical solution of the \textit{Falkner-Skan equation}. A posteriori error estimation governs the adaptivity of the mesh, specifically the…
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…
The convergence and optimality of adaptive mixed finite element methods for the Poisson equation are established in this paper. The main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure…
We develop all of the components needed to construct an adaptive finite element code that can be used to approximate fractional partial differential equations, on non-trivial domains in $d\geq 1$ dimensions. Our main approach consists of…
In this paper, we study an adaptive finite element method for a class of a nonlinear eigenvalue problems that may be of nonconvex energy functional and consider its applications to quantum chemistry. We prove the convergence of adaptive…
The paper is concerned with the adaptive finite element solution of linear elliptic differential equations using equidistributing meshes. A strategy is developed for defining this type of mesh based on residual-based a posteriori error…
The convergence of an adaptive mixed finite element method for general second order linear elliptic problems defined on simply connected bounded polygonal domains is analyzed in this paper. The main difficulties in the analysis are posed by…
This work reviews goal-oriented a posteriori error control, adaptivity and solver control for finite element approximations to boundary and initial-boundary value problems for stationary and non-stationary partial differential equations,…
We propose and analyze reliable and efficient a posteriori error estimators for an optimal control problem that involves a nondifferentiable cost functional, the Poisson problem as state equation and control constraints. To approximate the…
Stochastic models of chemical systems are often analysed by solving the corresponding Fokker-Planck equation which is a drift-diffusion partial differential equation for the probability distribution function. Efficient numerical solution of…
In this paper, we analyze the convergence and optimality of a standard adaptive nonconforming linear element method for the Stokes problem. After establishing a special quasi--orthogonality property for both the velocity and the pressure in…
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 propose a novel finite element method scheme for singularly perturbed advection-diffusion-reaction problems, which combines certain quantum-assisted stabilization scheme with a classical h-adaptive approach to provide automatic error…