Maria Lukácová-Medvid'ová

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.

Evolution Galerkin methods for hyperbolic systems in two space dimensions

, The subject of the paper is the analysis of three new evolution Galerkin schemes for a system of hyperbolic equations, and particularly for the wave equation system. The aim is to construct methods which take into account all of the infinitely many directions of propagation of bicharacteristics. The main idea of the evolution Galerkin methods is the following: the initial function is evolved using the characteristic cone and then projected onto a finite element space. A numerical comparison is given of the new methods with already existing methods, both those based on the use of bicharacteristics as well as commonly used finite difference and finite volume methods. We discuss the stability properties of the schemes and derive error estimates.

Well-balanced finite volume evolution Galerkin methods for the shallow water equations

We present a new well-balanced finite volume method within the framework of the finite volume evolution Galerkin (FVEG) schemes. The methodology will be illustrated for the shallow water equations with source terms modelling the bottom topography and Coriolis forces. Results can be generalized to more complex systems of balance laws. The FVEG methods couple a finite volume formulation with approximate evolution operators. The latter are constructed using the bicharacteristics of multidimensional hyperbolic systems, such that all of the infinitely many directions of wave propagation are taken into account explicitly. We derive a well-balanced approximation of the integral equations and prove that the FVEG scheme is well-balanced for the stationary steady states as well as for the steady jets in the rotational frame. Several numerical experiments for stationary and quasi-stationary states as well as for steady jets confirm the reliability of the well-balanced FVEG scheme.

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.

Error Estimates for a Combined Finite Volume--Finite Element Method for Nonlinear Convection--Diffusion Problems

The subject of this paper is the analysis of error estimates of the combined finite volume--finite element (FV--FE) method for the numerical solution of 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 finite elements. Under the assumption that the exact solution possesses some regularity properties and the triangulations are of a weakly acute type, with the aid of the discrete maximum principle and a priori estimates, error estimates of the method are proved.

Finite Volume Evolution Galerkin Methods for Hyperbolic Systems

The subject of the paper is the derivation and analysis of new multidimensional, high-resolution, finite volume evolution Galerkin (FVEG) schemes for systems of nonlinear hyperbolic conservation laws. Our approach couples a finite volume formulation with approximate evolution operators. The latter are constructed using the bicharacteristics of the multidimensional hyperbolic system such that all of the infinitely many directions of wave propagation are taken into account. In particular, we propose a new FVEG scheme, which is designed in such a way that for a linear wave equation system the approximate evolution operator calculates any one-dimensional planar wave exactly. This operator improves the stability of the FVEG scheme considerably, leading to a stability limit closer to 1. Using the results obtained for the wave equation system, a new approximate evolution operator for the linearized Euler equations is also derived. The integrals over the cell interfaces also need to be approximated with care; in this case, our choice of Simpsons rule is guided by stability analysis of model problems. Second order resolution is obtained by means of a piecewise bilinear recovery. Numerical experiments confirm the accuracy and multidimensional behavior of the new scheme.

On the Stability of Evolution Galerkin Schemes Applied to a Two-Dimensional Wave Equation System

The subject of the paper is the analysis of stability of the evolution Galerkin (EG) methods for the two-dimensional wave equation system. We apply von Neumann analysis and use the Fourier transformation to estimate the stability limits of both the first and the second order EG methods.

Error analysis of finite element and finite volume methods for some viscoelastic fluids

Abstract We present the error analysis of a particular Oldroyd-B type model with the limiting Weissenberg number going to infinity. Assuming a suitable regularity of the exact solution we study the error estimates of a standard finite element method and of a combined finite element/finite volume method. Our theoretical result shows first order convergence of the finite element method and the error of the order 𝓞(h3/4) for the finite element/finite volume method. These error estimates are compared and confirmed by the numerical experiments.

An application of 3-D kinematical conservation laws: propagation of a 3-D wavefront

Three-dimensional (3-D) kinematical conservation laws (KCL) are equations of evolution of a propagating surface

Finite volume evolution Galerkin methods for nonlinear hyperbolic systems

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