scholarly journals The lumped mass finite element method for a parabolic problem

1985 ◽  
Vol 26 (3) ◽  
pp. 329-354 ◽  
Author(s):  
C. M. Chen ◽  
V. Thomée
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Heat Equation ◽  
Galerkin Method ◽  
Parabolic Problem ◽  
Piecewise Linear ◽  
Optimal Order ◽  
Lumped Mass ◽  
Two Space Dimensions ◽  
Standard Galerkin Method

AbstractFor the heat equation in two space dimensions we consider semidiscrete and totally discrete variants of the lumped mass modification of the standard Galerkin method, using piecewise linear approximating functions, and demonstrate error estimates of optimal order in L2 and of almost optimal order in L∞.

On Preservation of Positivity in Some Finite Element Methods for the Heat Equation

2015 ◽  
Vol 15 (4) ◽  
pp. 417-437 ◽  
Author(s):  
Panagiotis Chatzipantelidis ◽  
Zoltan Horváth ◽  
Vidar Thomée
Keyword(s):  
Finite Element ◽  
Heat Equation ◽  
Piecewise Linear ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Dirichlet Boundary Conditions ◽  
Lumped Mass ◽  
The Maximum Principle ◽  
Standard Galerkin Method ◽  
Initial Boundary Value

AbstractWe consider the initial boundary value problem for the homogeneous heat equation, with homogeneous Dirichlet boundary conditions. By the maximum principle the solution is nonnegative for positive time if the initial data are nonnegative. We complement in a number of ways earlier studies of the possible extension of this fact to spatially semidiscrete and fully discrete piecewise linear finite element discretizations, based on the standard Galerkin method, the lumped mass method, and the finite volume element method. We also provide numerical examples that illustrate our findings.

scholarly journals Discrete Mixed Petrov-Galerkin Finite Element Method for a Fourth-Order Two-Point Boundary Value Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-18
Author(s):  
L. Jones Tarcius Doss ◽  
A. P. Nandini
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Fourth Order ◽  
Piecewise Linear ◽  
Quadrature Rule ◽  
Optimal Order ◽  
Galerkin Finite Element Method ◽  
Galerkin Finite Element ◽  
Splitting Technique ◽  
Element Method

A quadrature-based mixed Petrov-Galerkin finite element method is applied to a fourth-order linear ordinary differential equation. After employing a splitting technique, a cubic spline trial space and a piecewise linear test space are considered in the method. The integrals are then replaced by the Gauss quadrature rule in the formulation itself. Optimal ordera priorierror estimates are obtained without any restriction on the mesh.

scholarly journals A Numerical Method for SolvingTwo-Dimensional Nonlinear Parabolic ProblemsBased on a Preconditioning Operator

2020 ◽  
Vol 25 (4) ◽  
pp. 531-545
Author(s):  
Amir Hossein Salehi Shayegan ◽  
Ali Zakeri ◽  
Seyed Mohammad Hosseini
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Parabolic Problem ◽  
Time Variable ◽  
Two Dimensional ◽  
Helmholtz Operator ◽  
Mesh Free ◽  
Nonlinear Parabolic ◽  
Spline Finite Element ◽  
Mesh Free Method

This article considers a nonlinear system of elliptic problems, which is obtained by discretizing the time variable of a two-dimensional nonlinear parabolic problem. Since the system consists of ill-conditioned problems, therefore a stabilized, mesh-free method is proposed. The method is based on coupling the preconditioned Sobolev space gradient method and WEB-spline finite element method with Helmholtz operator as a preconditioner. The convergence and error analysis of the method are given. Finally, a numerical example is solved by this preconditioner to show the efficiency and accuracy of the proposed methods.

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