Characteristic-Galerkin and Galerkin/least-squares space-time formulations for the advection-diffusion equation with time-dependent domains

Abstract For the advection-diffusion equation, the characteristic-Galerkin formulations are obtained by temporal discretization of the total derivative. These formulations, by construction, are Eulerian-Lagrangian, and therefore can handle time-dependent domains without difficulty. The Galerkin/least-squares space-time formulation, on the other hand, is written over the space-time domain of a problem, and therefore can handle time-dependent domains with no implementational difficulty. The purpose of this paper is to compare these two formulations based on error estimates and numerical performance, in the context of the advection-diffusion equation.

[1]  O. Zienkiewicz.  The Finite Element Method In Engineering Science , 1971 .

[2]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: VI. Convergence analysis of the generalized SUPG formulation for linear time-dependent multi-dimensional advective-diffusive systems , 1987 .

[3]  O. Pironneau,et al.  Finite Elements and characteristics for some parabolic-hyperbolic problems , 1983 .

[4]  Richard H. Gallagher,et al.  Finite elements in fluids , 1975 .

[5]  T. Tezduyar,et al.  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 , 1992 .

[6]  A. Majda,et al.  Rates of convergence for viscous splitting of the Navier-Stokes equations , 1981 .

[7]  J. P. Benque,et al.  A new finite element method for Navier-Stokes equations coupled with a temperature equation , 1982 .

[8]  Einar M. Rønquist,et al.  An Operator-integration-factor splitting method for time-dependent problems: Application to incompressible fluid flow , 1990 .

[9]  Claes Johnson,et al.  A new approach to algorithms for convection problems which are based on exact transport + projection , 1992 .

[10]  T. F. Russell,et al.  NUMERICAL METHODS FOR CONVECTION-DOMINATED DIFFUSION PROBLEMS BASED ON COMBINING THE METHOD OF CHARACTERISTICS WITH FINITE ELEMENT OR FINITE DIFFERENCE PROCEDURES* , 1982 .

[11]  ChorinAlexandre Joel.  Numerical study of slightly viscous flow , 1989 .

[12]  A. Stroud.  Approximate calculation of multiple integrals , 1973 .

[13]  O. Pironneau.  On the transport-diffusion algorithm and its applications to the Navier-Stokes equations , 1982 .

[14]  J. Z. Zhu,et al.  The finite element method , 1977 .

[15]  Kenneth Eriksson,et al.  Adaptive streamline diffusion finite element methods for stationary convection-diffusion problems , 1993 .

[16]  O. Pironneau.  Finite Element Methods for Fluids , 1990 .

[17]  S. Mittal,et al.  A new strategy for finite element computations involving moving boundaries and interfaces—the deforming-spatial-domain/space-time procedure. II: Computation of free-surface flows, two-liquid flows, and flows with drifting cylinders , 1992 .