Related papers: Convergence and Optimality of Adaptive Mixed Finit…
An adaptive moving mesh finite element method is studied for the numerical solution of the porous medium equation with and without variable exponents and absorption. The method is based on the so-called moving mesh partial differential…
This paper studies the convergence difficulty of cohesive zone models in static analysis. It is shown that an inappropriate starting point of iterations in the Newton-Raphson method is responsible for the convergence difficulty. A simple,…
We construct a finite element method (FEM) for the infinity Laplacian. Solutions of this problem may be singular, which has prompted us to conduct an a posteriori analysis of the method deriving residual based estimators to drive an…
This paper develops an enhanced finite element method for approximating a class of variational problems which exhibit the \textit{Lavrentiev gap phenomenon} in the sense that the minimum values of the energy functional have a nontrivial gap…
This article examines the Dirichlet boundary control problem governed by the Poisson equation, where the control variables are square integrable functions defined on the boundary of a two dimensional bounded, convex, polygonal domain. It…
We propose a new practical adaptive refinement strategy for $hp$-finite element approximations of elliptic problems. Following recent theoretical developments in polynomial-degree-robust a posteriori error analysis, we solve two types of…
Mixed finite element methods are considered for a ferrofluid flow model with magnetization paralleled to the magnetic field. The ferrofluid model is a coupled system of the Maxwell equations and the incompressible Navier-Stokes equations.…
We propose a generalized finite element method for the strongly damped wave equation with highly varying coefficients. The proposed method is based on the localized orthogonal decomposition introduced and is designed to handle independent…
The fourth-order PDE that models the density variation of smectic A liquid crystals presents unique challenges in its (numerical) analysis beyond more common fourth-order operators, such as the classical biharmonic. While the operator is…
This research rigorously investigates the convergence of adaptive finite element methods for regularized variational models of quasi-static brittle fracture in elastic solids. We specifically examine a novel Ambrosio-Tortorelli (AT1)…
This paper presents a new parameter free partially penalized immersed finite element method and convergence analysis for solving second order elliptic interface problems. A lifting operator is introduced on interface edges to ensure the…
This work develops a convergence theory for H(div)-conforming finite element methods applied to the steady Oseen problem, focusing on cases where the exact finite element complex holds while the commuting diagram property may fail. The…
This paper is concerned with the analysis and implementation of robust finite element approximation methods for mixed formulations of linear elasticity problems where the elastic solid is almost incompressible. Several novel a posteriori…
This is a survey on the theory of adaptive finite element methods (AFEMs), which are fundamental in modern computational science and engineering but whose mathematical assessment is a formidable challenge. We present a self-contained and…
An improved finite difference method with compact correction term is proposed to solve the Poisson equations. The compact correction term is developed by a coupled high-order compact and low-order classical finite difference formulations.…
A low-order finite element method is constructed and analysed for an incompressible non-Newtonian flow problem with power-law rheology. The method is based on a continuous piecewise linear approximation of the velocity field and piecewise…
We consider mechanics of composite materials in which thin inclusions are modeled by lower-dimensional manifolds. By successively applying the dimensional reduction to junctions and intersections within the material, a geometry of…
Mesh adaptivity is a useful tool for efficient solution to partial differential equations in very complex geometries. In the present paper we discuss the use of polygonal mesh refinement in order to tackle two common issues: first,…
In this paper, we want to clarify the Gibbs phenomenon when continuous and discontinuous finite elements are used to approximate discontinuous or nearly discontinuous PDE solutions from the approximation point of view. For a simple step…
In this paper, we introduce new stable mixed finite elements of any order on polytopal mesh for solving second order elliptic problem. We establish optimal order error estimates for velocity and super convergence for pressure. Numerical…