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].