Weighted Extended B-Spline Approximation of Dirichlet Problems

Abstract

We describe a new finite element method which uses weighted extended B-splines on a regular grid as basis functions for solving Dirichlet problems on bounded domains in arbitrary dimensions. This web-method does not require any grid generation and can be implemented very efficiently. It yields smooth, high order accurate approximations with relatively low dimensional subspaces.AMS subject classifications. 65N30, 65N12, 41A63, 41A15 PII. S0036142900373208 1. Introduction. In engineering applications, the description of the geometry and the mesh generation process are often bottlenecks in finite element approximations of elliptic boundary value problems. While meshing is not easy, but fairly well understood for planar domains (see, e.g., [HL88]), it becomes a really challenging task in higher dimensions; see, e.g., [GT94,Fuc99]. Actually, the required computation times can be long compared to those for setting up and solving the linear system. Moreover, on unstructured meshes, usually linear or multilinear basis functions have to be used, since higher order approximations lead to huge linear systems. Therefore, some efforts have been made to develop meshless methods; see, e.g., the survey [BKO + 96] and the references therein. A central problem of meshless Galerkin methods is to incorporate boundary conditions of Dirichlet type. Apart from methods geared to special cases, the two basic concepts employed so far are due to Babuska [Bab73a,Bab73b]. While the penalty method [Bab73b, Arn82, BG98] incorporates the boundary conditions into the variation functional, the Lagrange multiplier technique [Bab73a, Bra81, Ste95] treats them as a constraint in the minimization. The theory for either method is not straightforward, and both approaches reveal slight drawbacks.• Both methods generate approximations which, in general, do not satisfy the boundary conditions exactly. Moreover, basis functions with very small support in the domain can lead to excessively large condition numbers of the Galerkin system. For this reason, nodal representations like those described in [BLG94, DT96, BM97] are in general preferred to regularly structured grids. • The Lagrange multiplier method does not lead to positive definite systems and, in its original form, requires via the LBB condition [Pit80, BF91] an undesirable correlation between the interior and the boundary mesh width [DK98]. A stabilized, but more involved variant is proposed in [Ste95].

Composite finite elements for 3D image based computing

“…Below we describe how this scaling is performed in the case of mass and stiffness matrices. The entries of the local matrix are weighted and accumulated into the global one (lines [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23].…”

“…In the vicinity of the interface or the boundary, the standard basis is modified to resolve either the shape of the domain boundary or to meet the jump condition imposed by the material coefficient. In contrast to the web-spline approach [22], the CFE method can be used both for problems with discontinuous coefficients and for problems with a complicated domain boundary.…”

Introduction to the Web-method and its applications

“…Equivalently, the coefficients e i,k in definition (3) have to be chosen so that all weighted polynomials wp, degree p n, can be written as linear combinations of the Web-splines B i . We state without proof the proper formula, which was derived in [12]. We select for each outer index j an array of inner indices closest to j (cf.…”

“…Since wv ∈ H n+1 does not imply that v ∈ H n+1 , even if w is smooth, a subtle analysis of the quotient u/w is necessary. For smooth weight functions of order 1 it was shown in [10,12] that…”

“…In contrast, the Web-method, introduced in [12,13,15], is a meshless finite element approximation which stays within the standard Ritz-Galerkin framework, proposing a new type of finite element subspace V h . The basis functions B i are weighted extended B-splines (Web-splines) on a tensor product grid and meet all basic finite element requirements.…”