A new extrapolation cascadic multigrid method for 3D elliptic boundary value problems on rectangular domains

In this paper, we develop a new extrapolation cascadic multigrid (ECMG$_{jcg}$) method, which makes it possible to solve 3D elliptic boundary value problems on rectangular domains of over 100 million unknowns on a desktop computer in minutes. First, by combining Richardson extrapolation and tri-quadratic Serendipity interpolation techniques, we introduce a new extrapolation formula to provide a good initial guess for the iterative solution on the next finer grid, which is a third order approximation to the finite element (FE) solution. And the resulting large sparse linear system from the FE discretization is then solved by the Jacobi-preconditioned Conjugate Gradient (JCG) method. Additionally, instead of performing a fixed number of iterations as cascadic multigrid (CMG) methods, a relative residual stopping criterion is used in iterative solvers, which enables us to obtain conveniently the numerical solution with the desired accuracy. Moreover, a simple Richardson extrapolation is used to cheaply get a fourth order approximate solution on the entire fine grid. Test results are reported to show that ECMG$_{jcg}$ has much better efficiency compared to the classical MG methods. Since the initial guess for the iterative solution is a quite good approximation to the FE solution, numerical results show that only few number of iterations are required on the finest grid for ECMG$_{jcg}$ with an appropriate tolerance of the relative residual to achieve full second order accuracy, which is particularly important when solving large systems of equations and can greatly reduce the computational cost. It should be pointed out that when the tolerance becomes smaller, ECMG$_{jcg}$ still needs only few iterations to obtain fourth order extrapolated solution on each grid, except on the finest grid. Finally, we present the reason why our ECMG algorithms are so highly efficient for solving such problems.Comment: 20 pages, 4 figures, 10 tables; abbreviated abstrac

A POSTERIORI ERROR ESTIMATES INCLUDING ALGEBRAIC ERROR: COMPUTABLE UPPER BOUNDS AND STOPPING CRITERIA FOR ITERATIVE SOLVERS

Abstract. We consider the finite volume and the lowest-order mixed finite element discretizations of a second-order elliptic pure diffusion model problem. The first goal of this paper is to derive guaranteed and fully computable a posteriori error estimates which take into account an inexact solution of the associated linear algebraic system. We show that the algebraic error can be simply bounded using the algebraic residual vector. Much better results are, however, obtained using the complementary energy of an equilibrated Raviart–Thomas–Nédélec discrete vector field whose divergence is given by a proper weighting of the residual vector. The second goal of this paper is to construct efficient stopping criteria for iterative solvers such as the conjugate gradients, GMRES, or Bi-CGStab. We claim that the discretization error, implied by the given numerical method, and the algebraic one should be in balance, or, more precisely, that it is enough to solve the linear algebraic system to the accuracy which guarantees that the algebraic part of the error does not contribute significantly to the whole error. Our estimates allow a reliable and cheap comparison of the discretization and algebraic errors. One can thus use them to stop the iterative algebraic solver at the desired accuracy level, without performing an excessive number of unnecessary additional iterations. Under the assumption of the relative balance between the two errors, we also prove the efficiency of our a posteriori estimates, i.e., we show that they also represent a lower bound, up to a generic constant, for the overall energy error. A local version of this result is also stated. Several numerical experiments illustrate the theoretical results. Key words. Second-order elliptic partial differential equation, finite volume method, mixed finite element method, a posteriori error estimates, iterative methods for linear algebraic systems, stopping criteria. AMS subject classifications. 65N15, 65N30, 76M12, 65N22, 65F1

The convergence of the cascadic conjugate-gradient method applied to elliptic problems in domains with re-entrant corners

We study the convergence properties of the cascadic conjugate-gradient method (CCG-method) which is a simplyfied version of the classical multigrid method without coarse-grid correction Nevertheless, the CCG-method converges with a rate which is independent of the number of unknowns and the number of grid levels. Here, we prove this property for two-dimensional elliptic second-order Dirichlet problems in a polygonal domain with an interior angle greater than #pi#. For piecewise linear finite elements we construct special nested triangulations which satisfy the conditions of a 'triangulation of type (h, #gamma#, L)' in the sense of I. Babuaka, R. Kellogg and J. Pitkaeranta. In this way we can guarantee both the same order of accuracy of the discrete solution and the same convergence rate of the CCG-method as in the case of quasiuniform triangulations of a convex polygonal domain. (orig.)SIGLEAvailable from TIB Hannover: RR 4487(1997,37) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDeutsche Forschungsgemeinschaft (DFG), Bonn (Germany)DEGerman