Hengguang Li

Uniform convergence of the multigrid V-cycle on graded meshes for corner singularities

SUMMARY This paper analyzes a multigrid (MG) V-cycle scheme for solving the discretized 2D Poisson equation with corner singularities. Using weighted Sobolev spaces K m a () and a space decomposition based on elliptic projections, we prove that the MG V-cycle with standard smoothers (Richardson, weighted Jacobi, Gauss– Seidel, etc.) and piecewise linear interpolation converges uniformly for the linear systems obtained by finite element discretization of the Poisson equation on graded meshes. In addition, we provide numerical experiments to demonstrate the optimality of the proposed approach. Copyright q 2008 John Wiley & Sons, Ltd.

Finite element analysis for the axisymmetric Laplace operator on polygonal domains

Abstract Let L ≔ − r − 2 ( r ∂ r ) 2 − ∂ z 2 . We consider the equation L u = f on a bounded polygonal domain with suitable boundary conditions, derived from the three-dimensional axisymmetric Poisson’s equation. We establish the well-posedness, regularity, and Fredholm results in weighted Sobolev spaces, for possible singular solutions caused by the singular coefficient of the operator L , as r → 0 , and by non-smooth points on the boundary of the domain. In particular, our estimates show that there is no loss of regularity of the solution in these weighted Sobolev spaces. Besides, by analyzing the convergence property of the finite element solution, we provide a construction of improved graded meshes, such that the quasi-optimal convergence rate can be recovered on piecewise linear functions for singular solutions. The introduction of a new projection operator from the weighted space to the finite element subspace, certain scaling arguments, and a calculation of the index of the Fredholm operator, together with our regularity results, are the ingredients of the finite element estimates.

A-priori analysis and the finite element method for a class of degenerate elliptic equations

Consider the degenerate elliptic operator L δ := -∂ 2 x - δ 2 x 2 ∂ 2 y on Ω:= (0,1) × (0,l), for δ > 0,1 > 0. We prove well-posedness and regularity results for the degenerate elliptic equation L δ u = f in Ω, u|∂Ω = 0 using weighted Sobolev spaces K m a . In particular, by a proper choice of the parameters in the weighted Sobolev spaces K m a , we establish the existence and uniqueness of the solution. In addition, we show that there is no loss of K m a -regularity for the solution of the equation. We then provide an explicit construction of a sequence of finite dimensional subspaces V n for the finite element method, such that the optimal convergence rate is attained for the finite element solution u n ∈ V n , i.e., ∥u - u n ∥ H1(Ω) < Cdim(V n )-m 2 ∥f∥ Hm-1(Ω) with C independent of / and n.

On Stability, Accuracy, and Fast Solvers for Finite Element Approximations of the Axisymmetric Stokes Problem by Hood-Taylor Elements

We provide a proof of both the stability and the approximation property for the finite element approximations of the axisymmetric Stokes problem by continuous piecewise polynomials of degree

Multigrid methods for saddle point problems: Stokes and Lamé systems

\kappa+1

A posteriori error estimation of hierarchical type for the Schrödinger operator with inverse square potential

for the velocity and continuous piecewise polynomials of degree

The effect of numerical integration on the finite element approximation of linear functionals

\kappa

The ¹_{} stability of the Ritz projection on graded meshes

for the pressure with any

RBF-PU method for pricing options under the jump–diffusion model with local volatility

\kappa\geq1

An anisotropic finite element method on polyhedral domains: interpolation error analysis

. New techniques are designed so that in this perspective, by a simple transformation, the existing theory developed in three dimensional Cartesian coordinates can be effectively exploited. In fact, this perspective provides a new way of developing theories for the axisymmetric Stokes problems and it can be applied potentially to other problems as well. A simple illustration is provided for the application in the development and analysis of fast solvers for the resulting discrete saddle point problems. Sample numerical experiments have been presented as well to confirm the theoretical results.