Mirko Rokyta

Convergence of upwind finite volume schemes for scalar conservation laws in two dimensions

This paper proves the convergence of a general class of monotone finite volume methods for numerical schemes of scalar conservation laws in two dimensions on unstructured meshes. There are convergence results for fractional step methods on cartesian grids and for finite element algorithms on unstructured grids. Even for finite volume methods there are some recent results concerning the Lax–Friedrichs and the Godunov finite volume method. The proof in this paper considers a general class including the Lax–Friedrichs and the Engquist–Osher finite volume schemes, and uses a completely different idea than in previous papers to control the entropy dissipation.

Finite-element solution of flow problems with trailing conditions

Abstract The paper deals with the finite-element solution of stream function problems describing nonviscous subsonic irrotational flows past profiles. The main emphasis is laid on the treatment of the nonstandard trailing stagnation conditions which lead to physically admissible solutions. The paper presents a general conception of stream function finite-element modelling of complicated flow problems and a complete theory of the finite-element approximations, including the investigation of the existence and uniqueness of the solution of the nonsymmetric discrete problem and the convergence of approximate solutions to the exact solution.

Error estimates for higher-order finite volume schemes for convection–diffusion problems

Abstract It is still an open problem to prove a priori error estimates for finite volume schemes of higher order MUSCL type, including limiters, on unstructured meshes, which show some improvement compared to first order schemes. In this paper we use these higher order schemes for the discretization of convection dominated elliptic problems in a convex bounded domain Ω in ℝ2 and we can prove such kind of an a priori error estimate. In the part of the estimate, which refers to the discretization of the convective term, we gain h1/2. Although the original problem is linear, the numerical problem becomes nonlinear, due to MUSCL type reconstruction/limiter technique.