## Abstract

We introduce a method for approximating essential boundary conditions-conditions of Dirichlet type-within the generalized finite element method (GFEM) framework. Our results apply to general elliptic boundary value problems of the form - ∑_{i, j = 1}^{n} (a^{ij} u_{xi})_{xj} + ∑_{i = 1}^{n} b^{i} u_{xi} + cu = f in Ω, u = 0 on ∂ Ω, where Ω is a smooth bounded domain. As test-trial spaces, we consider sequences of GFEM spaces, { S_{μ} }_{μ ≥ 1}, which are nonconforming (that is S_{μ} ⊄ H_{0}^{1} (Ω)). We assume that ∥ v ∥_{L2 (∂ Ω)} ≤ Ch_{μ}^{m} ∥ v ∥_{H1 (Ω)}, for all v ∈ S_{μ}, and there exists u_{I} ∈ S_{μ} such that ∥ u - u_{I} ∥_{H1 (Ω)} ≤ Ch_{μ}^{j} ∥ u ∥_{Hj + 1 (Ω)}, 0 ≤ j ≤ m, where u ∈ H^{m + 1} (Ω) is the exact solution, m is the expected order of approximation, and h_{μ} is the typical size of the elements defining S_{μ}. Under these conditions, we prove quasi-optimal rates of convergence for the GFEM approximating sequence u_{μ} ∈ S_{μ} of u. Next, we extend our analysis to the inhomogeneous boundary value problem - ∑_{i, j = 1}^{n} (a^{ij} u_{xi})_{xj} + ∑_{i = 1}^{n} b^{i} u_{xi} + cu = f in Ω, u = g on ∂ Ω. Finally, we outline the construction of a sequence of GFEM spaces S_{μ} ⊂ over(S, ̃)_{μ}, μ = 1, 2, ..., that satisfies our assumptions.

Original language | English (US) |
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Pages (from-to) | 175-183 |

Number of pages | 9 |

Journal | Journal of Computational and Applied Mathematics |

Volume | 218 |

Issue number | 1 |

DOIs | |

State | Published - Aug 15 2008 |

## All Science Journal Classification (ASJC) codes

- Computational Mathematics
- Applied Mathematics