This paper is concerned with a specific finite element strategy for solving elliptic boundary value problems in domains with corners and edges. First, the anisotropic singular behaviour of the solution is described. Then the finite element method with anisotropic, graded meshes and piecewise linear shape functions is investigated for such problems; the schemes exhibit optimal convergence rates ...

Kleitman and Rothschild (Trans. Amer. Math. Soc. 205 (1975), 205-220) gave an asymptotic formula for the number of partial orders with ground-set In]. We give a shorter proof of their result and extend it to count the number of pairs (P, -<), where P is a partial order on [hi and -< is a linear extension of P. This gives us an asymptotic formula for (a) the average number of linear extensions o...

Abstract. In this work, we study the finite element approximation of a model optimal control problem governed by the system describing the two-phase incompressible flow in porous media, with the aim to maximize production of oil from petroleum reservoirs. We first give the proof for the existence of the solutions of the control problem. The optimality conditions are then obtained and the existe...

This paper proposes and analyzes two fully discrete mixed interior penalty discontinuous Galerkin (DG) methods for the fourth order nonlinear Cahn-Hilliard equation. Both methods use the backward Euler method for time discretization and interior penalty discontinuous Galerkin methods for spatial discretization. They differ from each other on how the nonlinear term is treated, one of them is bas...

R. Rivest has recently proposed the following intriguing conjecture: Let x* denote an arbitrary fixed element in an n-element partially ordered set P, and for each k in {1, 2, , n let Nk be the number of order-preserving maps from P onto {1, 2,..., n} that map x* into k. Then the sequence N1,'' ', N,, is unimodal. This note proves the conjecture for the special case in which P can be covered by...

A spectral/hp element discretisation permits both geometric flexibility and beneficial convergence properties to be attained simultaneously. The choice of elemental polynomial order has a profound effect on the efficiency of different implementation strategies with their performance varying substantially for low and high order spectral/hp discretisations. We examine how careful selection of the...