Petrov-Galerkin Approximation Research Articles

Numerical Schemes for the Hamilton–Jacobi and Level Set Equations on Triangulated Domains

Borrowing from techniques developed for conservation law equations, numerical schemes which discretize the Hamilton–Jacobi (H-J), level set, and Eikonal equations on triangulated domains are presented. The first scheme is a provably monotone discretization for the H-J equations. Unfortunately, the basic scheme lacks Lipschitz continuity of the numerical Hamiltonian. By employing a “virtual” edge flipping technique, local Lipschitz continuity of the numerical flux is restored on acute triangulations. Next, schemes are introduced and developed based on the weaker concept of positive coefficient approximations for homogeneous Hamiltonians. These schemes possess a discrete maximum principle on arbitrary triangulations and exhibit local Lipschitz continuity of the numerical Hamiltonian. Finally, a class of Petrov–Galerkin approximations is considered. These schemes are stabilized via a least-squares bilinear form. The Petrov–Galerkin schemes do not possess a discrete maximum principle but generalize to high order accuracy. Discretization of the level set equation also requires the numerical approximation of a mean curvature term. A simple mass-lumped Galerkin approximation is presented in Section 6 and analyzed using maximum principle analysis. The use of unstructured meshes permits several forms of mesh adaptation which have been incorporated into numerical examples. These numerical examples include discretizations of convex and nonconvex forms of the H-J equation, the Eikonal equation, and the level set equation.

Finite element analysis of transient creep problems

Stabilized mixed Petrov-Galerkin (Galerkin/least-squares) and unstable Galerkin finite element approximations of transient elasto-creep problems in stress-velocity formulation are analyzed. The asymptotic behaviour of both continuum and discrete problems are discussed and the main results on the numerical analysis and error estimates are presented. Even with the same order of interpolation for velocity and stress fields, the Petrov—Galerkin approximation is uniformly convergent and tends asymptotically to the corresponding steady state solution with the expected exponential rate while the same does not happen to the Galerkin approximation.

Optimal error estimation for Petrov-Galerkin methods in two dimensions

We examine the optimality of conforming Petrov-Galerkin approximations for the linear convection-diffusion equation in two dimensions. Our analysis is based on the Riesz representation theorem and it provides an optimal error estimate involving the smallest possible constantC. It also identifies an optimal test space, for any choice of consistent norm, as that whose image under the Riesz representation operator is the trial space. By using the Helmholtz decomposition of the Hilbert space [L 2(Ω)]2, we produce a construction for the constantC in which the Riesz representation operator is not required explicitly. We apply the technique to the analysis of the Galerkin approximation and of an upwind finite element method.

Pointwise Accuracy of a Stable Petrov–Galerkin Approximation to the Stokes Problem

The finite-element approximation of the Stokes problem due to Hughes, Franca, and Balestra [Comput. Meth. App!. Mech. Engrg., 59 (1986), pp. 85–99] is analyzed in maximum norm. The method consists of modifying the usual bilinear form associated with the saddle-point structure to become coercive over the finite-element space. Exploiting the enhanced stability—thus getting rid of the inf sup condition—yields quasi-optimal $L^\infty $-error estimates for both velocity and pressure.