Ramon Codina

Comparison of some finite element methods for solving the diffusion-convection-reaction equation

Abstract In this paper we describe several finite element methods for solving the diffusion-convection-reaction equation. None of them is new, although the presentation is non-standard in an effort to emphasize the similarities and differences between them. In particular, it is shown that the classical SUPG method is very similar to an explicit version of the Characteristic-Galerkin method, whereas the Taylor-Galerkin method has a stabilization effect similar to a sub-grid scale model, which is in turn related to the introduction of bubble functions.

Stabilized finite element approximation of transient incompressible flows using orthogonal subscales

Abstract In this paper we present a stabilized finite element method to solve the transient Navier–Stokes equations based on the decomposition of the unknowns into resolvable and subgrid scales. The latter are approximately accounted for, so as to end up with a stable finite element problem which, in particular, allows to deal with convection-dominated flows and the use of equal velocity–pressure interpolations. Three main issues are addressed. The first is a method to estimate the behavior of the stabilization parameters based on a Fourier analysis of the problem for the subscales. Secondly, the way to deal with transient problems discretized using a finite difference scheme is discussed. Finally, the treatment of the nonlinear term is also analyzed. A very important feature of this work is that the subgrid scales are taken as orthogonal to the finite element space. In the transient case, this simplifies considerably the numerical scheme.

Stabilization of incompressibility and convection through orthogonal sub-scales in finite element methods

Two apparently different forms of dealing with the numerical instability due to the incompressibility constraint of the Stokes problem are analyzed in this paper. The first of them is the stabilization through the pressure gradient projection, which consists of adding a certain least-squares form of the difference between the pressure gradient and its L2 projection onto the discrete velocity space in the variational equations of the problem. The second is a sub-grid scale method, whose stabilization effect is very similar to that of the Galerkin/least-squares (GLS) method for the Stokes problem. It is shown here that the first method can also be recast in the framework of sub-grid scale methods with a particular choice for the space of sub-scales. This leads to a new stabilization procedure, whose applicability to stabilize convection is also studied in this paper.

A discontinuity-capturing crosswind-dissipation for the finite element solution of the convection-diffusion equation

Abstract To avoid the local oscillations that still remain using the streamline-upwind/Petrov-Galerkin formulation for the scalar convection-diffusion equation, the introduction of a nonlinear crosswind dissipation is proposed. It is shown that the method is less overdiffusive than other discontinuity-capturing techniques and has better numerical behavior. The design of the crosswind diffusion is based on the study of the discrete maximum principle for some simple cases.

A stabilized finite element method for generalized stationary incompressible flows

Abstract In this paper, we describe a finite element formulation for the numerical solution of the stationary incompressible Navier–Stokes equations including Coriolis forces and the permeability of the medium. The stabilized method is based on the algebraic version of the sub-grid scale approach. We first describe this technique for general systems of convection–diffusion–reaction equations and then we apply it to the linearized flow equations. The important point is the design of the matrix of stabilization parameters that the method has. This design is based on the identification of the stability problems of the Galerkin method and a scaling of variables argument to determine which coefficients must be included in the stabilization matrix. This, together with the convergence analysis of the linearized problem, leads to a simple expression for the stabilization parameters in the general situation considered in the paper. The numerical analysis of the linearized problem also shows that the method has optimal convergence properties.

The characteristic‐based‐split procedure: an efficient and accurate algorithm for fluid problems

In 1995 the two senior authors of the present paper introduced a new algorithm designed to replace the Taylor–Galerkin (or Lax–Wendroff) methods, used by them so far in the solution of compressible flow problems. The new algorithm was applicable to a wide variety of situations, including fully incompressible flows and shallow water equations, as well as supersonic and hypersonic situations, and has proved to be always at least as accurate as other algorithms currently used. The algorithm is based on the solution of conservation equations of fluid mechanics to avoid any possibility of spurious solutions that may otherwise result. The main aspect of the procedure is to split the equations into two parts, (1) a part that is a set of simple scalar equations of convective–diffusion type for which it is well known that the characteristic Galerkin procedure yields an optimal solution; and (2) the part where the equations are self-adjoint and therefore discretized optimally by the Galerkin procedure. It is possible to solve both the first and second parts of the system explicitly, retaining there the time step limitations of the Taylor–Galerkin procedure. But it is also possible to use semi-implicit processes where in the first part we use a much bigger time step generally governed by the Peclet number of the system while the second part is solved implicitly and is unconditionally stable. It turns out that the characteristic-based-split (CBS) process allows equal interpolation to be used for all system variables without difficulties when the incompressible or nearly incompressible stage is reached. It is hoped that the paper will help to make the algorithm more widely available and understood by the profession and that its advantages can be widely realised. Copyright

On stabilized finite element methods for linear systems of convection-diffusion-reaction equations

Abstract A stabilized finite element method for solving systems of convection–diffusion-reaction equations is studied in this paper. The method is based on the subgrid scale approach and an algebraic approximation to the subscales. After presenting the formulation of the method, it is analyzed how it behaves under changes of variables, showing that it relies on the law of change of the matrix of stabilization parameters associated to the method. An expression for this matrix is proposed for the case of general coupled systems of equations that is an extension of the expression proposed for a one-dimensional (1D) model problem. Applications of the stabilization technique to the Stokes problem with convection and to the bending of Reissner–Mindlin plates are discussed next. The design of the matrix of stabilization parameters is based on the identification of the stability deficiencies of the standard Galerkin method applied to these two problems.

A finite element formulation for the Stokes problem allowing equal velocity-pressure interpolation

Abstract In this paper we study a variational formulation of the Stokes problem that accommodates the use of equal velocity-pressure finite element interpolations. The motivation of this method relies on the analysis of a class of fractional-step methods for the Navier-Stokes equations for which it is known that equal interpolations yield good numerical results. The reason for this turns out to be the difference between two discrete Laplacian operators computed in a different manner. The formulation of the Stokes problem considered here aims to reproduce this effect. From the analysis of the finite element approximation of the problem we obtain stability and optimal error estimates using velocity-pressure interpolations satisfying a compatibility condition much weaker than the inf-sup condition of the standard formulation. In particular, this condition is fulfilled by the most common equal order interpolations.

Stabilized finite element method for the transient Navier–Stokes equations based on a pressure gradient projection

Abstract In this paper we present a stabilized finite element formulation for the transient incompressible Navier–Stokes equations. The main idea is to introduce as a new unknown of the problem the projection of the pressure gradient onto the velocity space and to add to the incompresibility equation the difference between the Laplacian of the pressure and the divergence of this new vector field. This leads to a pressure stabilization effect that allows the use of equal interpolation for both velocities and pressures. In the case of the transient equations, we consider the possibility of treating the pressure gradient projection either implicitly or explicity. In the first case, the number of unknowns of the problem is substantially increased with respect to the standard Galerkin formulation. Nevertheless, iterative techniques may be used in order to uncouple the calculation of the pressure gradient projection from the rest of unknowns (velocity and pressure). When this vector field is treated explicitly, the increment of computational cost of the stabilized formulation with respect to the Galerkin method is very low. We provide a stability estimate for the case of the simple backward Euler time integration scheme for both the implicit and the explicit treatment of the pressure gradient projection.

A general algorithm for compressible and incompressible flows. Part III: The semi‐implicit form

SUMMARY In this paper we consider some particular aspects related to the semi-implicit version of a fractional step finite element method for compressible flows that we have developed recently. The first is the imposition of boundary conditions. We show that no boundary conditions at all need to be imposed in the first step where an intermediate momentum is computed. This allows us to impose the real boundary conditions for the pressure, a point that turns out to be very important for compressible flows. The main difficulty of the semi-implicit form of the scheme arises in the solution of the continuity equation, since it involves both the density and the pressure. These two variables can be related through the equation of state, which in turn introduces the temperature as a variable in many cases. We discuss here the choice of variables (pressure or density) and some strategies to solve the continuity equation. The final point that we study is the behaviour of the scheme in the incompressible limit. It is shown that the method has an inherent pressure dissipation that allows us to reach this limit without having to satisfy the classical compatibility conditions for the interpolation of the velocity and the pressure. # 1998 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids, 27: 13‐32 (1998)