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In this paper, we will discuss the geometric-based algebraic multigrid (AMG) method for two-dimensional linear elasticity problems discretized using quadratic and cubic elements. First, a two-level method is proposed by analyzing the relationship between the linear finite element space and higher-order finite element space. And then a geometric-based AMG method is obtained with the existing solver used as a solver on the first coarse level. The resulting AMG method is applied to some typical elasticity problems including the plane strain problem with jumps in Young's modulus. The results of various numerical experiments show that the proposed AMG method is much more robust and efficient than a classical AMG solver that is applied directly to the high-order systems alone. Moreover, we present the corresponding theoretical analysis for the convergence of the proposed AMG algorithms. These theoretical results are also confirmed by some numerical tests. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper formulates, analyses and discusses 2D and 3D static and dynamic model problems in the orthopaedic practice. Finite element approximations, algorithms and iterative methods for constrained optimization are discussed. Since the conjugate gradient method is one of the most effective methods for both unconstrained and constrained optimization, it can be applied without or with preconditioning for solving the basic step of the discretized problem. A comparison of several preconditioned conjugate gradient methods is discussed. The problems discussed are applied to analyses of real patients. Finally, the the numerical results are discussed. Copyright © 1999 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper presents a general method to associate the operator splitting for the Peaceman-Rachford procedure with a decomposition of the domain in problems arising from finite element discretization of partial differential equations. The algorithm is provably convergent without any symmetry requirement. Moreover, this method possesses the significant advantage of making the linear systems of the Peaceman-Rachford iteration block diagonal and therefore perfectly appropriate for parallel processing. Not only is sparsity not affected but a reduction of the bandwidth occurs. In fact, for appropriate choices of nonconforming finite element spaces, this method makes directly possible elementwise processing. This option remains available in general for higher-dimensional problems by applying the splitting algorithm recursively. Practical implementation requires nothing more than the standard finite element assembly procedure and some bookkeeping to relate a few different orderings of the nodes. In addition to all these attractive features, the method is rapidly convergent and remains highly competitive even when used on a serial machine. [ABSTRACT FROM AUTHOR]

The FETI method with the natural coarse grid is combined with the penalty method to develop an efficient solver for elliptic variational inequalities. A proof is given that a prescribed bound on the norm of feasibility of solution may be achieved with a value of the penalty parameter that does not depend on the discretization parameter and that an approximate solution with the prescribed bound on violation of the Karush–Kuhn–Tucker conditions may be found in a number of steps that does not depend on the discretization parameter. Results of numerical experiments with parallel solution of a model problem discretized by up to more than eight million of nodal variables are in agreement with the theory and demonstrate numerically both optimality of the penalty and scalability of the algorithm presented. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We present experiments with various solvers for large sparse generalized symmetric matrix eigenvalue problems. These problems occur in the computation of a few of the lowest frequencies of standing electromagnetic waves in resonant cavities with the finite element method. The solvers investigated are (1) subspace iteration, (2) block Lanczos algorithm, (3) implicitly restarted Lanczos algorithm and (4) Jacobi–Davidson algorithm. The experiments have been conducted on a Hewlett-Packard Exemplar S-Class system. Copyright © 1999 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]