Jiří Felcman

On the Convergence of a Combined Finite Volume{Finite Element Method for Nonlinear Convection{Diffusion Problems

We present the convergence analysis of an efficient numerical method for the solution of an initial-boundary value problem for a scalar nonlinear conservation law equation with a diffusion term. Nonlinear convective terms are approximated with the aid of a monotone finite volume scheme considered over the finite volume mesh dual to a triangular grid, whereas the diffusion term is discretized by piecewise linear conforming triangular elements. Under the assumption that the triangulations are of weakly acute type, with the aid of the discrete maximum principle, a priori estimates, and some compactness arguments based on the use of the Fourier transform with respect to time, the convergence of the approximate solutions to the exact solution is proved, provided that the mesh size tends to zero. c 1997 John Wiley & Sons, Inc.

Combined finite element-finite volume solution of compressible flow

Abstract The paper is concerned with numerical modelling of inviscid as well as viscous gas flow. The method is based on upwind flux vector splitting finite volume schemes on various types of unstructured grids. In the case of viscous flow we apply a combined method using the finite volume scheme for the discretization of inviscid first order terms of the system and the finite element approximation of viscous dissipative terms. Special attention is paid to higher order schemes and suitable adaptive strategy for a precise resolution of shock waves. Moreover, we summarize the convergence results obtained for a model nonlinear scalar conservation law equation with a diffusion term. Some computational results are presented. In this paper only two-dimensional flow is treated, but the extension to the three-dimensional case is possible.

ANALYSIS OF A COMBINED BARYCENTRIC FINITE VOLUME—NONCONFORMING FINITE ELEMENT METHOD FOR NONLINEAR CONVECTION-DIFFUSION PROBLEMS

We present the convergence analysis of an efficient numerical method for the solution of an initial-boundary value problem for a scalar nonlinear conservation law equation with a diffusion term. Nonlinear convective terms are approximated with the aid of a monotone finite volume scheme considered over the finite volume barycentric mesh, whereas the diffusion term is discretized by piecewise linear nonconforming triangular finite elements. Under the assumption that the triangulations are of weakly acute type, with the aid of the discrete maximum principle, a priori estimates and some compactness arguments based on the use of the Fourier transform with respect to time, the convergence of the approximate solutions to the exact solution is proved, provided the mesh size tends to zero.

On the discrete Friedrichs inequality for nonconforming finite elements

In this paper we establish the validity of the discrete Friedrichs inequality for piecewise linear Crouzeix-Raviart nonconforming finite elements in polygonal domains. It represents an extension of an important result proven for a polygonal convex domain to a general polygonal nonconvex domain. This result has applications in the analysis of exterior approximations of partial differential equations as, e.g., the Navier-Stokes equations and convection-diffusion problems.

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.

Grid refinement/alignment in 3D flow computations

The paper is concerned with the grid refinement/coarsening/alignment technique for the adaptive solution of the 3D compressible flow. The necessary condition for the properties of the tetrahedrization on which the discretization error is below the prescribed tolerance is formulated. The interpolation error in the density and in the Mach number for the inviscid and viscous flow, respectively, is used to control this necessary condition. The original smoothing procedure for the generally discontinuous approximate solution is proposed in terms of edge based Hesse matrices. This allows to define the concept of the optimal mesh and to adopt the standard anisotropic mesh adaptation technique for the equidistribution of the interpolation error function. Geometrically interpreted, the optimal mesh is almost equilateral tetrahedrization with respect to the solution dependent Riemann norm of edges. For its construction, the iterative algorithm is proposed. The theoretical considerations are completed by the numerical example of the adaptive solution of the 3D channel flow.

Adaptive Methods in Internal and External Flow Computations

To illustrate the application of mathematics to industrial problems we will discuss a mathematical model for the two-dimensional inviscid as well as viscous transonic gas flow in channels, past airfoils and cascades of profiles.