Special finite element methods for a class of second order elliptic problems with rough coefficients. (English) Zbl 0807.65114

Babuška, Ivo; Caloz, Gabriel; Osborn, John E.

SIAM J. Numer. Anal. 31, No. 4, 945-981 (1994).

The authors consider elliptic Dirichlet boundary value problems of the form \[ -\text{div}(a(x,y)\nabla u(x,y))= f(x,y)\quad\text{in }\Omega\subset \mathbb{R}^ 2,\quad u=0\quad\text{on }\partial\Omega\tag{1} \] with some coefficient \(a(.,.)\) sharply varying in at most one direction parallel to one axis either of the Cartesian (straight line unidirectional) or of the polar (curvilinear unidirectional) coordinate system. Such problems arise in the cases of straight line unidirectional or tubular (curvilinear unidirectional) composites.
The solution of the boundary value problem (1) will also be rough. The standard finite element approach needs a fine mesh in order to fit the interfaces with the finite element edges. The authors propose three methods for constructing special trial and test spaces such that the \(O(h)\) accuracy in \(H^ 1\) is achieved as in the \(H^ 2\)-regularity case although it is not necessary to fit the interfaces with the finite elements.

Reviewer: U.Langer (Linz)

MSC:

65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15	Error bounds for boundary value problems involving PDEs
65N50	Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
35J25	Boundary value problems for second-order elliptic equations

Keywords:

second order elliptic problems; rough coefficients; boundary value problem; finite elements