A Crank-Nicolson type space-time finite element method for computing on moving meshes
type: Item
instance of: scholarly article
title: A Crank-Nicolson type space-time finite element method for computing on moving meshes
zbMATH Open document ID: 0961.65091
author: Peter Hansbo
published in: Journal of Computational Physics
publication date: 2000-11-23T00:00:00Z
review text: A variant of the continuous finite element method suitable for computation on deforming meshes is introduced. The method is an extension of the Crank-Nicolson method, and computational experience confirms that the method retains the second-order temporal accuracy of the Crank-Nicolson method. The exact conditions necessary for second order accuracy with regards to the deformation of the mesh remains to be investigated. The interest in space-time finite element methods for flows in deforming domains is motivated by the fact that it allows for mimicking the use of Lagrangian coordinates without introducing actual differencing along the characteristics. This means that if the velocity field used to convect the mesh is different from the actual velocity field (which is the usual situation), there will remain a small residual convective term. The resulting nonsymmetry of the corresponding matrix problem is the price paid for consistency; the suggested method will work also if it is difficult to align the mesh along the characteristics. Furthermore, by coupling the approximation of the solution to the approximation of the flow field, there is an immediate gain when the method is applied to nonlinear flow problems where the flow field is identical to the solution, as, e.g., in the Navier-Stokes equations. Clearly, in such a case the residual convective term becomes identical to zero by definition, which automatically reduces (the discretized) Navier-Stokes equation to the Stokes equation without further approximations.
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 76D05
Mathematics Subject Classification ID: 65M50
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 76M10
zbMATH DE Number: 1443372
zbMATH Keywords: space-time finite element method
zbMATH Keywords: Crank-Nicolson method
zbMATH Keywords: evolution problems
zbMATH Keywords: moving meshes
zbMATH Keywords: Navier-Stokes equations
reviewed by: Leonid B. Chubarov
MaRDI profile type: Publication
full work available at URL: jcph.2000.6436
OpenAlex ID: W1998526580
cites work: Continuous Finite Elements in Space and Time for the Heat Equation
cites work: A second order finite element method for the one-dimensional Stefan problem
cites work: Numerical computation of the free boundary for the two-dimensional Stefan problem by space-time finite elements
cites work: Q5690463
cites work: The characteristic streamline diffusion method for convection-diffusion problems
cites work: The characteristic streamline diffusion method for the time-dependent incompressible Navier-Stokes equations
cites work: Lagrangian incompressible flow computations in three dimensions by use of space‐time finite elements
cites work: Moving finite element methods by use of space-time elements: I. Scalar problems
cites work: Finite elements and characteristics applied to advection-diffusion equations
cites work: One-Step Piecewise Polynomial Galerkin Methods for Initial Value Problems
cites work: Galerkin-Type Approximations which are Discontinuous in Time for Parabolic Equations in a Variable Domain
cites work: Stability and Convergence of a Generalized Crank-Nicolson Scheme on a Variable Mesh for the Heat Equation
cites work: A new approach to algorithms for convection problems which are based on exact transport + projection
cites work: Generalised Galerkin methods for hyperbolic problems
cites work: On the transport-diffusion algorithm and its applications to the Navier-Stokes equations
cites work: Characteristic-Galerkin and Galerkin/least-squares space-time formulations for the advection-diffusion equation with time-dependent domains
cites work: A new strategy for finite element computations involving moving boundaries and interfaces --- The deforming-spatial-domain/space-time procedure. I: The concept and the preliminary numerical tests
cites work: A Stable Finite Element Method for Initial-Boundary Value Problems for First-Order Hyperbolic Systems
cites work: Q3875490
DOI: 10.1006/JCPH.2000.6436
description: scientific article; zbMATH DE number 1443372