Inf-sup testing of upwind methods

Abstract

SUMMARYWe propose inf-sup testing for ÿnite element methods with upwinding used to solve convection-di usion problems. The testing evaluates the stability of a method and compactly displays the numerical behaviour as the convection e ects increase. Four discretization schemes are considered: the standard Galerkin procedure, the full upwind method, the Galerkin least-squares scheme and a high-order derivative artiÿcial di usion method. The study shows that, as expected, the standard Galerkin method does not pass the inf-sup tests, whereas the other three methods pass the tests. Of these methods, the high-order derivative artiÿcial di usion procedure introduces the least amount of artiÿcial di usion.

Pseudo-divergence-free element free Galerkin method for incompressible fluid flow

“…Finally, replacing the definition of U(x) and a(x) given by (12) and (18) in (17), one gets the final expression, which completes the proof,…”

“…Recall that each component of U(x) depends on the corresponding exact derivatives of u, see (12). Now, observe that each component of a(x) shall depend on the corresponding pseudo-derivatives; that is, for |k| = 0, ..., m…”

“…If this is not the case, alternative formulations (usually depending on a numerical parameter) are devised to circumvent the LBB condition and enable the use of velocity-pressure pairs that are unstable in the standard Galerkin formulation. Finally, note that it is not trivial to verify analytically the LBB condition for a given interpolation of velocity and pressure, and this has spurred the use of numerical inf-sup testing [9][10][11][12].…”

The inf–sup condition and its evaluation for mixed finite element methods

“…Hence, mixed ®nite element formulations can now be evaluated as shown in this paper before it is claimed that an effective formulation has been developed. Inf±sup test results for one-dimensional high P eclet number¯ow problem [6]. The convection-diusion problem corresponds to a constant prescribed velocity throughout the domain and prescribed temperatures at both ends of the domain.…”

“…Then, as derived in Ref. [6], the inf± sup value c h is obtained by solving the following eigenvalue problem for the smallest eigenvalue k min , A…”