A novel heuristic residual analysis is proposed to derive a computationally cost-eeective residual projection operator in multigrid with the ve-point Red-Black Gauss-Seidel relaxation for solving the two dimensional Poisson equation. This optimal residual injection operator is as cheap as the trivial injection operator, but is more eecient than the costly full-weighting operator and achieves ne...

Mixed nite element application to solve the Stokes equations using di erent smoothers by mean of multigrid technique has been investigated. Multigrid technique's e ciency for positive de nite linear systems is proved but it is much more challenging in the case of saddle-point problems. The e ciency of a multigrid method is highly dependent on the smoother and the coarse grid solver. A two-dimen...

Let A be a matrix such that the diagonal matrix D with the same diagonal as A is invertible. It is well known that if (1) A satisfies the Sassenfeld condition then its Gauss-Seidel scheme is convergent, and (2) if D−1A certifies certain classical diagonal dominance conditions then the Jacobi iterations for A are convergent. In this paper we generalize the second result and extend the first resu...

The Gauss–Seidel projection method (GSPM) (Wang et al., J. Comp. Phys., vol. 171, pp. 357–372, 2001) is a simple, efficient, and unconditionally stable method for micromagnetics simulations. We present an improvement of the method for small values of the damping parameter. With the new method, we are able to carry out fully resolved simulations of the magnetization reversal process in the prese...

In this paper, we study preconditioning techniques for both H(grad) and H(curl) problems. For an H(grad) elliptic problem discretized by high-order finite elements with a hierachical basis of k-th order, we design and analyse an parallel AMG preconditioner based on a two-level method and a block Gauss–Seidel smoothing technique. For an H(curl) elliptic problem discretized by high-order edge fin...

A convergence theory is presented for approximations of continuous-time optimal control problems based on a Gauss pseudospectral discretization. Under assumptions of coercivity and smoothness, the Gauss pseudospectral method has a local minimizer that converges exponentially fast in the sup-norm to a local minimizer of the continuous-time optimal control problem. The convergence theorem is pres...

This work wishes to support various mathematical issues concerning the iterative methods with the help of new programming languages. We consider a way to show how problems in math have an answer by using different academic resources and different thoughts. Here we treat methods like Gauss-Seidel’s, Cramer’s and Gauss-Jordan’s.

The RK5GL3 method is a numerical method for solving initial value problems in ordinary differential equations, and is based on a combination of a fifth-order Runge-Kutta method and 3-point Gauss-Legendre quadrature. In this paper we describe the propagation of local errors in this method, and show that the global order of RK5GL3 is expected to be six, one better than the underlying RungeKutta m...

The variable-step Störmer-Cowell integrator is a non-summed, double-integration multi-step integrator derived in variable-step form. The method has been implemented with a Shampine-Gordon style error control algorithm that uses an approximation of the local error at each step to choose the step size for the subsequent step. In this paper, the variable-step Störmer-Cowell method is compared to s...