# The International Conference on Computational Mathematics ICCM-2004

## Abstracts

Numerical solution of differential and integral equations

## P-multigrid for hp-discretization of the Euler and Navier-Stokes equations

### Feodoritova O.B. , Zhukov V.T. , Young D. P.

Keldysh Institute of Applied Mathematics of RAS (Moscow)

A two-level iterative algorithm is presented for solution of linear systems of equations arising from higher order discretizations of PDEs. The 2D compressible Euler and Navier-Stokes equations discretized by FEM on unstructured triangular grids are considered. Steady-state solutions of these nonlinear equations are computed by the implicit time-steping scheme [1]. The two-level algorithm is studied as a solver for the linear systems Au = b that have to be solved in each time step. Here u denotes the coefficient vector of unknown solution with respect to a finite element basis, A is the stiffness matrix. The choice of a FE basis affects numerical solution of linear systems. We investigate convergence rate of the algorithm for the lagrangean(L) and hierarchical(H) bases, usually exploited in FE application. In the L-basis each local element function is a polynomial of the same order p. The H-basis uses the standard linear element modes which are supplemented hierarchical polynomials of higher degrees up to p. Computational cost of linear system solution of depends usually on mesh parameter h and polynomial order p of finite elements. To achieve order independent (or weekly dependent) property we elaborate the algorithm, given here. It is presented p-stage of the full hp-multigrid that is under development. The results of numerical experiments for a few problems (convection-diffusion, Euler and Navier-Stokes equations) are given.

1. V. Venkatakrishnan, S. Allmaras, D. Kamenetskii, F. Johnson. Higher Order Schemes for the Compressible Navier-Stokes Equations. AIAA-2003-3987 (2003).

2. A.A. Martynov and S.Yu. Medvedev. emph{A robust method of anisotropic grid generation. Grid Generation: theory and applications. Computing Centre RAS, Moscow, 2002,c. 266-275

Note. Abstracts are published in author's edition

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