Superconvergence in H1-norm of a difference finite element method for the heat equation in a 3D spatial domain with almost-uniform mesh

2020 ◽  
Vol 86 (1) ◽  
pp. 357-395
Author(s):  
Xinlong Feng ◽  
Ruijian He ◽  
Zhangxin Chen
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Heat Equation ◽  
Spatial Domain ◽  
Uniform Mesh ◽  
Element Method

A Lagrangian Uniform-Mesh Finite Element Method Applied to Problems Governed by Poisson’s Equation

2007 ◽  
Author(s):  
Linxia Gu ◽  
Ashok V. Kumar
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Weak Form ◽  
Lagrangian Formulation ◽  
Uniform Mesh ◽  
Interpolation Function ◽  
Step Functions ◽  
Surface Integration ◽  
Interpolation Functions ◽  
Element Method

A method is presented for the solution of Poisson’s Equations using a Lagrangian formulation. The interpolation functions are the Lagrangian operation of those used in the classical finite element method, which automatically satisfy boundary conditions exactly even though there are no nodes on the boundaries of the domain. The integration is introduced in an implicit way by using approximated step functions. Classical surface integration terms used in the weak form are unnecessary due to the interpolation function in the Lagrangian formulation. Furthermore, the Lagrangian formulation simplified the connection between the mesh and the solid structures, thus providing a very easy way to solve the problems without a conforming mesh.

Weak Galerkin mixed finite element method for heat equation

2018 ◽  
Vol 123 ◽  
pp. 180-199 ◽  
Author(s):  
Chenguang Zhou ◽  
Yongkui Zou ◽  
Shimin Chai ◽  
Qian Zhang ◽  
Hongze Zhu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Heat Equation ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Weak Galerkin ◽  
Element Method

PARAMETER-UNIFORM FINITE ELEMENT METHOD FOR TWO-PARAMETER SINGULARLY PERTURBED PARABOLIC REACTION-DIFFUSION PROBLEMS

2012 ◽  
Vol 09 (04) ◽  
pp. 1250047 ◽  
Author(s):  
M. K. KADALBAJOO ◽  
ARJUN SINGH YADAW
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Reaction Diffusion ◽  
Rectangular Domain ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
Spatial Direction ◽  
Small Parameters ◽  
Diffusion Problems ◽  
Element Method

In this paper, parameter-uniform numerical methods for a class of singularly perturbed one-dimensional parabolic reaction-diffusion problems with two small parameters on a rectangular domain are studied. Parameter-explicit theoretical bounds on the derivatives of the solutions are derived. The method comprises a standard implicit finite difference scheme to discretize in temporal direction on a uniform mesh by means of Rothe's method and finite element method in spatial direction on a piecewise uniform mesh of Shishkin type. The method is shown to be unconditionally stable and accurate of order O(N-2( ln N)2 + Δt). Numerical results are given to illustrate the parameter-uniform convergence of the numerical approximations.

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