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.

scholarly journals A Compact FEM Implementation for Parabolic Integro-Differential Equations in 2D

Algorithms ◽  
2020 ◽  
Vol 13 (10) ◽  
pp. 242
Author(s):  
Gujji Murali Mohan Reddy ◽  
Alan B. Seitenfuss ◽  
Débora de Oliveira Medeiros ◽  
Luca Meacci ◽  
Milton Assunção ◽  
...  
Keyword(s):  
Finite Element ◽  
Differential Equations ◽  
Piecewise Linear ◽  
Time Discretization ◽  
Comsol Multiphysics ◽  
Integral Term ◽  
Finite Element Software ◽  
Time Stepping ◽  
Backward Euler ◽  
Shaped Domain

Although two-dimensional (2D) parabolic integro-differential equations (PIDEs) arise in many physical contexts, there is no generally available software that is able to solve them numerically. To remedy this situation, in this article, we provide a compact implementation for solving 2D PIDEs using the finite element method (FEM) on unstructured grids. Piecewise linear finite element spaces on triangles are used for the space discretization, whereas the time discretization is based on the backward-Euler and the Crank–Nicolson methods. The quadrature rules for discretizing the Volterra integral term are chosen so as to be consistent with the time-stepping schemes; a more efficient version of the implementation that uses a vectorization technique in the assembly process is also presented. The compactness of the approach is demonstrated using the software Matrix Laboratory (MATLAB). The efficiency is demonstrated via a numerical example on an L-shaped domain, for which a comparison is possible against the commercially available finite element software COMSOL Multiphysics. Moreover, further consideration indicates that COMSOL Multiphysics cannot be directly applied to 2D PIDEs containing more complex kernels in the Volterra integral term, whereas our method can. Consequently, the subroutines we present constitute a valuable open and validated resource for solving more general 2D PIDEs.

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 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.

scholarly journals A Finite Element Splitting Method for a Convection-Diffusion Problem

2020 ◽  
Vol 20 (4) ◽  
pp. 717-725 ◽  
Author(s):  
Vidar Thomée
Keyword(s):  
Finite Element ◽  
Splitting Method ◽  
Diffusion Problem ◽  
Euler Method ◽  
Euler Approximation ◽  
Convection Diffusion ◽  
Time Stepping ◽  
Backward Euler ◽  
Spatially Periodic ◽  
Forward Euler

AbstractFor a spatially periodic convection-diffusion problem, we analyze a time stepping method based on Lie splitting of a spatially semidiscrete finite element solution on time steps of length k, using the backward Euler method for the diffusion part and a stabilized explicit forward Euler approximation on {m\geq 1} intervals of length {k/m} for the convection part. This complements earlier work on time splitting of the problem in a finite difference context.

scholarly journals A Banach space mixed formulation for the unsteady Brinkman–Forchheimer equations

2020 ◽  
Author(s):  
Sergio Caucao ◽  
Ivan Yotov
Keyword(s):  
Banach Space ◽  
Time Discretization ◽  
Monotone Operators ◽  
Rates Of Convergence ◽  
Mixed Formulation ◽  
Model Parameters ◽  
Weak Formulation ◽  
Finite Element Approximations ◽  
Backward Euler ◽  
Discrete Finite Element

Abstract We propose and analyse a mixed formulation for the Brinkman–Forchheimer equations for unsteady flows. Our approach is based on the introduction of a pseudostress tensor related to the velocity gradient and pressure, leading to a mixed formulation where the pseudostress tensor and the velocity are the main unknowns of the system. We establish existence and uniqueness of a solution to the weak formulation in a Banach space setting, employing classical results on nonlinear monotone operators and a regularization technique. We then present well posedness and error analysis for semidiscrete continuous-in-time and fully discrete finite element approximations on simplicial grids with spatial discretization based on the Raviart–Thomas spaces of degree $k$ for the pseudostress tensor and discontinuous piecewise polynomial elements of degree $k$ for the velocity and backward Euler time discretization. We provide several numerical results to confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method for a range of model parameters.

scholarly journals Goal-Oriented Error Estimation for the Fractional Step Theta Scheme

2014 ◽  
Vol 14 (2) ◽  
pp. 203-230 ◽  
Author(s):  
Dominik Meidner ◽  
Thomas Richter
Keyword(s):  
Time Discretization ◽  
Posteriori Error ◽  
Numerical Dissipation ◽  
Error Estimator ◽  
Fractional Step ◽  
Weighted Residual Method ◽  
Time Stepping ◽  
Backward Euler ◽  
Galerkin Formulation ◽  
Theta Method

Abstract. In this work, we derive a goal-oriented a posteriori error estimator for the error due to time-discretization of nonlinear parabolic partial differential equations by the fractional step theta method. This time-stepping scheme is assembled by three steps of the general theta method, that also unifies simple schemes like forward and backward Euler as well as the Crank–Nicolson method. Further, by combining three substeps of the theta time-stepping scheme, the fractional step theta time-stepping scheme is derived. It possesses highly desired stability and numerical dissipation properties and is second order accurate. The derived error estimator is based on a Petrov–Galerkin formulation that is up to a numerical quadrature error equivalent to the theta time-stepping scheme. The error estimator is assembled as one weighted residual term given by the dual weighted residual method and one additional residual estimating the Galerkin error between time-stepping scheme and Petrov–Galerkin formulation.

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 Controlling the dynamics of Burgers equation with a high-order nonlinearity

2004 ◽  
Vol 2004 (62) ◽  
pp. 3321-3332 ◽  
Author(s):  
Nejib Smaoui
Keyword(s):  
Steady State ◽  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Nonlinear Boundary ◽  
Burgers Equation ◽  
High Order ◽  
Time Stepping ◽  
Backward Euler ◽  
High Order Nonlinearity ◽  
Steady State Solutions

We investigate analytically as well as numerically Burgers equation with a high-order nonlinearity (i.e.,ut=νuxx−unux+mu+h(x)). We show existence of an absorbing ball inL2[0,1]and uniqueness of steady state solutions for all integern≥1. Then, we use an adaptive nonlinear boundary controller to show that it guarantees global asymptotic stability in time and convergence of the solution to the trivial solution. Numerical results using Chebychev collocation method with backward Euler time stepping scheme are presented for both the controlled and the uncontrolled equations illustrating the performance of the controller and supporting the analytical results.

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