Multigrid methods with web-splines

Abstract

We describe and analyze a multigrid algorithm for finite element approximations of second order elliptic boundary value problems with weighted extended b-splines (web-splines). This new technique provides high accuracy with relatively low-dimensional subspaces, does not require any grid generation, and is ideally suited for hierarchical solution techniques. In particular, we show that the standard W-cycle yields uniform convergence, i.e., the required number of iterations is bounded independent of the grid width.

A New Class of Stabilized WEB-Spline-Based Mesh-Free Finite Elements for the Approximation of Maxwell Equations

“…A central problem of mesh-less Galerkin method is to incorporate boundary conditions of Dirichlet type. The weighted extended B-spline approximation takes care of not only the boundary constraints but also the issue of well conditioning of the Galerkin systems [12,13]. This essential new feature of constructing well-conditioned basis is of cardinal importance in the this work.…”

“…Following the notations as in [7,12,13], we describe the procedure of constructing the WEB-splines and discuss the approximating properties of the WEB-space as a finite element space. For k ∈ 2 , and h > 0 define the scaled translates:…”

“…Weighted extended B-spline methods for boundary value problems are proposed as a natural generalization of standard B-splines to address two important issues of exact fulfillment of the boundary conditions and well conditioning of the Galerkin system. The development of this new strategy for the Laplace operator problem is due to Hollig et al in [12,13]. As will be described in the next section, a weight function, w, is being multiplied to the B-spline functions to ensure that all WEB splines satisfy the boundary conditions exactly.…”

Introduction to the Web-method and its applications

“…For example, it can be shown without too much effort that the w-cycle converges with a grid independent rate, even though the Web-spaces are not nested [14].…”

Programming finite element methods with weighted B-splines

“…Moreover, simple preconditioning procedures appear to be sufficient to solve the ill-conditioned finite element systems with acceptable accuracy. An application, for which stability seems indispensable, is multigrid methods [10]. So far it is still an open problem whether the extension procedure, required for theoretical purposes, is essential also for an optimal implementation.…”