We consider the numerical approximation of boundary layer phenomena occuring in many singularly perturbed problems in mechanics, such as plate and shell problems. We present guidelines for the e ective resolution of such l a yers in the context of exisiting, commercial p and hp nite element (FE) version codes. We show that if high order, \spectral" elements are available, then just two elements are su cient to approximate these layers at a near-exponential rate, independently of the problem parameters thickness or Reynolds number. We present hp mesh design principles for situations where both corner singularities and boundary layers are present.
The hp-version of the finite element method is applied to a singularly perturbed reaction-diffusion equation posed on an interval or a two-dimensional domain with an analytic boundary. On suitably designed Spectral Boundary Layer meshes, robust exponential convergence in a "balanced" norm is shown. This "balanced" norm is an ε-weighted H 1 -norm, where the weighting in terms of the singular perturbation parameter ε is such that, in contrast to the standard energy norm, boundary layer contributions do not vanish in the limit ε → 0. Robust exponential convergence in the maximum norm is also established. We illustrate the theoretical findings with two numerical experiments.
SUMMARYThe singular function boundary integral method is applied for the solution of a Laplace equation problem over an L-shaped domain. The solution is approximated by the leading terms of the local asymptotic solution expansion, while the Dirichlet boundary conditions are weakly enforced by means of Lagrange multipliers. Estimates of great accuracy are obtained for the leading singular coe cients, as well as for the Lagrange multipliers. Comparisons are made with recent numerical results in the literature.
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