Global Higher-Order Asymptotic Expansions for Anisotropic Singularly Perturbed Elliptic Equations
For an anisotropic singularly perturbed elliptic equation -div(A_epsilon grad u_epsilon)=f, whose diffusion matrix scales derivatives differently across two coordinate blocks, one seeks an approximation of u_epsilon valid over the whole domain rather than only on interior subdomains. This concept develops the arbitrary-order expansion u_epsilon = sum_k epsilon^k u_k in anisotropic Sobolev spaces, proving L^2-gradient convergence rates of order epsilon^d under regularity of the data and the block coefficients, in the regime where the lateral boundary layer vanishes.
Global asymptotic expansion for anisotropic singularly perturbed elliptic problems
This paper studies anisotropic singular perturbations of linear second-order elliptic boundary-value problems, in which a small parameter epsilon multiplies the derivatives in only some of the coordi…