Pavel B. Bochev

Stabilization of Low-order Mixed Finite Elements for the Stokes Equations

We present a new family of stabilized methods for the Stokes problem. The focus of the paper is on the lowest order velocity-pressure pairs. While not LBB compliant, their simplicity and attractive computational properties make these pairs a popular choice in engineering practice. Our stabilization approach is motivated by terms that characterize the LBB deficiency of the unstable spaces. The stabilized methods are defined by using these terms to modify the saddle-point Lagrangian associated with the Stokes equations. The new stabilized methods offer a number of attractive computational properties. In contrast to other stabilization procedures, they are parameter free, do not require calculation of higher order derivatives or edge-based data structures, and always lead to symmetric linear systems. Furthermore, the new methods are unconditionally stable, achieve optimal accuracy with respect to solution regularity, and have simple and straightforward implementations. We present numerical results in two and three dimensions that showcase the excellent stability and accuracy of the new methods.

Finite Element Methods of Least-Squares Type

We consider the application of least-squares variational principles to the numerical solution of partial differential equations. Our main focus is on the development of least-squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows, linear elasticity, and convection-diffusion. For many of these problems, least-squares principles offer numerous theoretical and computational advantages in the algorithmic design and implementation of corresponding finite element methods that are not present in standard Galerkin discretizations. Most notably, the use of least-squares principles leads to symmetric and positive definite algebraic problems and allows us to circumvent stability conditions such as the inf-sup condition arising in mixed methods for the Stokes and Navier--Stokes equations. As a result, application of least-squares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance.

Least-squares finite element methods

Least-squares finite element methods are an attractive class of methods for the numericalnsolution of partial differential equations. They are motivated by the desire to recover, inngeneral settings, the advantageous features of Rayleigh�Ritz methods such as the avoidance ofndiscrete compatibility conditions and the production of symmetric and positive definite discretensystems. The methods are based on the minimization of convex functionals that are constructednfrom equation residuals. This paper focuses on theoretical and practical aspects of least-squarenfinite element methods and includes discussions of what issues enter into their construction,nanalysis, and performance. It also includes a discussion of some open problems.

Analysis of least squares finite element methods for the Stokes equations

In this paper we consider the application of least-squares principles to the approximate solution of the Stokes equations cast into a first-order velocity-vorticity-pressure system. Among the most attractive features of the resulting methods are that the choice of approximating spaces is not subject to the LBB condition and a single continuous piecewise polynomial space can be used for the approximation of all unknowns, that the resulting discretized problems involve only symmetric, positive definite systems of algebraic equations, that no artificial boundary conditions for the vorticity need be devised, and that accurate approximations are obtained for all variables, including the vorticity. Here we study two classes of least-squares methods for the velocity-vorticity-pressure equations. The first one uses norms prescribed by the a priori estimates of Agmon, Douglis, and Nirenberg and can be analyzed in a completely standard manner. However, conforming discretizations of these methods require C continuity of the finite element spaces, thus negating the advantages of the velocity-vorticity-pressure fomulation. The second class uses weighted L-norms of the residuals to circumvent this flaw. For properly choosen mesh-dependent weights, it is shown that the approximations to the solutions of the Stokes equations are of optimal order. The results of some computational experiments are also provided; these illustrate, among other things, the necessity of introducing the weights. AMS Subject Classification: 65N30, 65N12, 76M10

Analysis of Velocity-Flux First-Order System Least-Squares Principles for the Navier--Stokes Equations: Part I

This paper develops a least-squares approach to the solution of the incompressible Navier--Stokes equations in primitive variables. As with our earlier work on Stokes equations, we recast the Navier--Stokes equations as a first-order system by introducing a velocity-flux variable and associated curl and trace equations. We show that a least-squares principle based on L2 norms applied to this system yields optimal discretization error estimates in the H1 norm in each variable, including the velocity flux. nAn analogous principle based on the use of an H-1 norm for the reduced system (with no curl or trace constraints) is shown to yield similar estimates, but now in the L2 norm for velocity-flux and pressure. Although the H-1 least-squares principle does not allow practical implementation, these results are critical to the analysis of a practical least-squares method for the reduced system based on a discrete equivalent of the negative norm. A practical method of this type is the subject of a companion paper. Finally, we establish optimal multigrid convergence estimates for the algebraic system resulting from the L2 norm approach.

Analysis of Least-Squares Finite Element Methods for the Navier--Stokes Equations

In this paper we study finite element methods of least-squares type for the stationary, incompressible Navier--Stokes equations in two and three dimensions. We consider methods based on velocity-vorticity-pressure form of the Navier--Stokes equations augmented with several nonstandard boundary conditions. Least-squares minimization principles for these boundary value problems are developed with the aid of the Agmon--Douglis--Nirenberg (ADN) elliptic theory. Among the main results of this paper are optimal error estimates for conforming finite element approximations and analysis of some nonstandard boundary conditions. Results of several computational experiments with least-squares methods which illustrate, among other things, the optimal convergence rates are also reported.

A Taxonomy of Consistently Stabilized Finite Element Methods for the Stokes Problem

Stabilized mixed methods can circumvent the restrictive inf-sup condition without introducing penalty errors. For properly chosen stabilization parameters these methods are well-posed for all conforming velocity-pressure pairs. However, their variational forms have widely varying properties. First, stabilization offers a choice between weakly or strongly coercive bilinear forms that give rise to linear systems with identical solutions but very different matrix properties. Second, coercivity may be conditional upon a proper choice of a stabilizing parameter. Here we focus on how these two aspects of stabilized methods affect their accuracy and efficient iterative solution. We present results that indicate a preference of Krylov subspace solvers for strongly coercive formulations. Stability criteria obtained by finite element and algebraic analyses are compared with numerical experiments. While for two popular classes of stabilized methods, sufficient stability bounds correlate well with numerical stability, our experiments indicate the intriguing possibility that the pressure-stabilized Galerkin method is unconditionally stable.

An Improved Algebraic Multigrid Method for Solving Maxwell's Equations

We propose two improvements to the Reitzinger and Schoberl algebraic multigrid (AMG) method for solving the eddy current approximations to Maxwells equations. The main focus in the Reitzinger/Schoberl method is to maintain null space properties of the weak

Least-squares methods for the velocity-pressure-stress formulation of the Stokes equations

nabla times nabla times

Accuracy of least-squares methods for the Navier-Stokes equations

operator on coarse grids. While these null space properties are critical, they are not enough to guarantee h-independent convergence of the overall multigrid method. We illustrate how the Reitzinger/Schoberl AMG method loses h-independence due to the somewhat limited approximation property of the grid transfer operators. We present two improvements to these operators that not only maintain the important null space properties on coarse grids but also yield significantly improved multigrid convergence rates. The first improvement is based on smoothing the Reitzinger/Schoberl grid transfer operators. The second improvement is obtained by using higher order nodal interpolation to derive the corresponding AMG interpolation operators. While not completely h-independent, the resulting AMG/CG method demonstrates improved convergence behavior while maintaining low operator complexity.