An algebraic multigrid method for linear elasticity. (English) Zbl 1163.65336
Summary: We present an algebraic multigrid (AMG) method for the efficient solution of linear block-systems stemming from a discretization of a system of partial differential equations (PDEs). It generalizes the classical AMG approach for scalar problems to systems of PDEs in a natural blockwise fashion. We apply this approach to linear elasticity and show that the block interpolation, described in this paper, reproduces the rigid body modes, i.e., the kernel elements of the discrete linear elasticity operator. It is well known from geometric multigrid methods that this reproduction of the kernel elements is an essential property to obtain convergence rates which are independent of the problem size. We furthermore present results of various numerical experiments in two and three dimensions. They confirm that the method is robust with respect to variations of the Poisson ratio \(\nu\). We obtain rates \(\rho<0.4\) for \(\nu<0.4\). These measured rates clearly show that the method provides fast convergence for a large variety of discretized elasticity problems.
MSC:
65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |
74G15 | Numerical approximation of solutions of equilibrium problems in solid mechanics |
65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |
65F10 | Iterative numerical methods for linear systems |
74B05 | Classical linear elasticity |