scholarly journals Resolvent Estimates in l_p for Discrete Laplacians on Irregular Meshes and Maximum-norm Stability of Parabolic Finite Difference Schemes

2001 ◽  
Vol 1 (1) ◽  
pp. 3-17 ◽  
Author(s):  
Michel Crouzeix ◽  
Vidar Thomée
Keyword(s):  
Finite Difference ◽  
Piecewise Linear ◽  
Maximum Norm ◽  
Parabolic Problems ◽  
Resolvent Estimate ◽  
Finite Difference Schemes ◽  
Weighted Norm ◽  
Resolvent Estimates ◽  
Lumped Mass ◽  
Mesh Ratio

AbstractIn an attempt to show maximum-norm stability and smoothing estimates for finite element discretizations of parabolic problems on nonquasi-uniform triangulations we consider the lumped mass method with piecewise linear finite elements in one and two space dimensions. By an energy argument we derive resolvent estimate for the associated discrete Laplacian, which is then a finite difference operator on an irregular mesh, which show that this generates an analytic semigroup in l_p for p‹∞ uniformly in the mesh, assuming in the two-dimensional case that the triangulations are of Delaunay type, and with a logarithmic bound for p=∞. By a different argument based on a weighted norm estimate for a discrete Green's function this is improved to hold without a logarithmic factor for p=∞ in one dimension under a weak mesh-ratio condition. Our estimates are applied to show stability also for time stepping methods.

Spectrally Consistent Approximations to the Matrix Exponent and Their Applications to Boundary Layer Problem

2002 ◽  
Vol 2 (4) ◽  
pp. 354-377
Author(s):  
Vladimir V. Bobkov
Keyword(s):  
Piecewise Linear ◽  
Parabolic Problems ◽  
Resolvent Estimate ◽  
Weighted Norm ◽  
Lumped Mass ◽  
Boundary Layer Problem ◽  
Lumped Mass Method ◽  
The Matrix ◽  
Mesh Ratio ◽  
Ratio Condition

AbstractIn an attempt to show maximum-norm stability and smoothing estimates for finite element discretizations of parabolic problems on nonquasi-uniform triangulations we consider the lumped mass method with piecewise linear finite elements in one and two space dimensions. By an energy argument we derive resolvent estimate for the associated discrete Laplacian, which is then a finite difference operator on an irregular mesh, which show that this generates an analytic semigroup in l_p for p‹∞ uniformly in the mesh, assuming in the two-dimensional case that the triangulations are of Delaunay type, and with a logarithmic bound for p=∞. By a different argument based on a weighted norm estimate for a discrete Green's function this is improved to hold without a logarithmic factor for p=∞ in one dimension under a weak mesh-ratio condition. Our estimates are applied to show stability also for time stepping methods.

scholarly journals On Positivity and Maximum-Norm Contractivity in Time Stepping Methods for Parabolic Equations

2010 ◽  
Vol 10 (4) ◽  
pp. 421-443 ◽  
Author(s):  
A.H. Schatz ◽  
V. Thomèe ◽  
L.B. Wahlbin
Keyword(s):  
Piecewise Linear ◽  
Small Time ◽  
Maximum Norm ◽  
Lumped Mass ◽  
Finite Element Approximations ◽  
Time Stepping ◽  
Backward Euler ◽  
Discrete Analogues ◽  
Carry Over ◽  
Standard Galerkin Method

Abstract In an earlier paper the last two authors studied spatially semidiscrete piecewise linear finite element approximations of the heat equation and showed that, in the case of the standard Galerkin method, the solution operator of the initial-value problem is neither positive nor contractive in the maximum-norm for small time, but that for the lumped mass method these properties hold, if the triangulations are essentially of Delaunay type. In this paper we continue the study by considering fully discrete analogues obtained by discretization also in time. The above properties then carry over to the backward Euler time stepping method, but for other methods the results are more restrictive. We discuss in particular the θ-method and the (0; 2) Padé approximation in one space dimension.

PARALLEL DOMAIN DECOMPOSITION METHODS WITH THE OVERLAPPING OF SUBDOMAINS FOR PARABOLIC PROBLEMS

1996 ◽  
Vol 06 (08) ◽  
pp. 1169-1185 ◽  
Author(s):  
GRIGORII I. SHISHKIN ◽  
PETR N. VABISHCHEVICH
Keyword(s):  
Finite Difference ◽  
Domain Decomposition ◽  
Domain Decomposition Method ◽  
Approximate Solutions ◽  
Decomposition Methods ◽  
Difference Schemes ◽  
Parabolic Problems ◽  
Uniform Grid ◽  
Finite Difference Schemes ◽  
Dimensional Boundary

For a model of two-dimensional boundary value problem for a second-order parabolic equation, finite difference schemes on the base of a domain decomposition method, oriented on modern parallel computers, is constructed. In the used finite difference schemes iterations at time levels are not applied; some subdomains overlap. We study two classes of schemes characterized by synchronous and asynchronous implementations. It is shown that, under refining grids, the approximate solutions do converge to the exact one in the uniform grid norm.

Monotone Finite Difference Schemes for Quasilinear Parabolic Problems with Mixed Boundary Conditions

2016 ◽  
Vol 16 (2) ◽  
pp. 231-243
Author(s):  
Francisco José Gaspar ◽  
Francisco Javier Lisbona ◽  
Piotr P. Matus ◽  
Vo Thi Kim Tuyen
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Mixed Boundary ◽  
Finite Difference Methods ◽  
Neumann Boundary ◽  
Numerical Approach ◽  
Parabolic Problems ◽  
Finite Difference Schemes ◽  
Quasilinear Parabolic Problems ◽  
Quasilinear Parabolic

AbstractIn this paper, we consider finite difference methods for two-dimensional quasilinear parabolic problems with mixed Dirichlet–Neumann boundary conditions. Some strong two-side estimates for the difference solution are provided and convergence results in the discrete norm are proved. Numerical examples illustrate the good performance of the proposed numerical approach.

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