Jakub Šístek

Accurate Solution of Corner Singularities in Axisymmetric and Plane Flows Using Adjusted Mesh of Finite Elements

We investigate the finite element solution of the flow of incompressible fluid in channels or in axisymmetric tubes with abrupt changes of diameters. The problems of singularities near the nonconvex corners are solved on a priori refined meshes. The refinement is based on a priori error estimate and on the asymptotic behaviour of the solution near the corners. In case of “geometric” singularities this is an efficient alternative to the adaptive mesh refinement. We present numerical results of flows in a channel with sharp obstacle, and in a channel with sharp cavity.

Finite Element Mesh Adjusted to Singularities Applied to Axisymmetric and Plane Flow

We consider the Navier-Stokes equations for incompressible flow in axisymmetric tubes with abrupt changes of diameter. This paper is based on results for the asymptotics of the solution in the vicinity of nonconvex internal angles where the velocities possess an expansion u(ρ, ϑ) = ρ γ ϕ(ϑ) + …, where ρ, ϑ are local spherical coordinates. Problems with corners, edges, etc. are often successfully solved by the finite element method using an adaptive strategy usually based on a posteriori error estimates. In our paper we suggest an alternative approach for the mesh refinement near the corners, which makes use of the above expansion. It gives very precise results in a cheap way. We give numerical results and show the pros and cons of this approach.

The Effect of Irregular Interfaces on the BDDC Method for the Navier-Stokes Equations

We investigate the effect of interface irregularity on the convergence of the BDDC method for Navier-Stokes equations. A benchmark problem of a sequence of contracting channels is proposed to evaluate the robustness of the iterative solver with respect to element aspect ratios at the interface. Partitioners based on graph of the mesh and the geometry of the domain are compared. It is shown, that the convergence is significantly improved by avoiding irregular interfaces for the benchmark problem as well as for an industrial problem of oil flow in hydrostatic bearing.

Accuracy Analysis Based on a Posteriori Error Estimates of SemiGLS Stabilization of FEM for Solving Navier-Stokes Equations

The accuracy of the stabilized finite element solution of incompressible flow problems with higher Reynolds numbers is studied. We use a modification of the Galerkin Least Squares Method called semiGLS. A posteriori error estimates are used as the principal tool for the accuracy analysis. The problem of singularities is considered. Numerical results are presented.