A Petrov-Galerkin discretization with optimal test space of a mild-weak formulation of convection-diffusion equations in mixed form. (English) Zbl 1311.65144
The authors study a variational formulation of second-order elliptic equations in mixed form that is obtained by piecewise integrating one of the two equations in the system w.r.t. a partition of the domain into mesh cells. A Petrov-Galerkin discretization with optimal test functions is applied. These optimal test functions can be computed by solving local problems. Well-posedness and optimal error estimates are proved. In the second part of the paper, the application to convection-diffusion problems is studied. The available freedom in the variational formulation and in its optimal Petrov-Galerkin discretization is used to construct a method that allows a passing to a converging method in the convective limit, being a necessary condition to retain convergence and having a bound on the cost for a vanishing diffusion. The theoretical findings are supported by several numerical results.
Reviewer: Abdallah Bradji (Annaba)
MSC:
65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
35J25 | Boundary value problems for second-order elliptic equations |
65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
65N15 | Error bounds for boundary value problems involving PDEs |