Localized-adjoint-finite-element-method for sub-grid stabilization of convection-dominated transport on a triangular mesh

Abstract

The advection-diffusion equation is notoriously difficult to solve for higher Peclet number when using standard Galerkin methods. Strong oscillations occur in regions of higher gradient. In order to improve the Galerkin solution two successful stabilized methods have been considered in the last decade, which are the Streamline Upwinding Petrov Galerkin method and the residual-free bubbles method. Moreover Herrera, in the context of his algebraic theory for boundary methods, has shown that optimal schemes can be derived by using optimal test functions satisfying a local adjoint boundary value problem. In this paper we apply Herrera's approach to consider unstructured triangular meshes. In order make the residual error vanish locally at each element an adjoint integro-differential boundary-value problem has been derived and solved under the hypothesis of dominant advection by the methods of successive approximations and multiple-scale perturbation. We have applied the proposed approach to the linear and quadratic elements, thereby showing that the stabilized quadratic Galerkin elements perform better than the stabilized linear Galerkin elements. Comparison with other stabilization methods is also illustrated.