Least-squares mixed finite element methods for non-selfadjoint elliptic problems: I. Error estimates

Abstract

A least-squares mixed finite element method for general secondorder non-selfadjoint elliptic problems in two-and three-dimensional domains is formulated and analyzed. The finite element spaces for the primary solution approximation u h and the flux approximation σ h consist of piecewise polynomials of degree k and r respectively. The method is mildly nonconforming on the boundary. The cases k = r and k + 1 = r are studied. It is proved that the method is not subject to the LBB-condition. Optimal L 2 -and H 1 -error estimates are derived for regular finite element partitions. Numerical experiments, confirming the theoretical rates of convergence, are presented.

Analysis of a non-standard mixed finite element method with applications to superconvergence

“…It is however not immediately clear what its merits are. The coercivity constant is now equal to one and not to 1 − γ as we saw in (18). This is in principle an advantage.…”

Weak coupling of solutions of first-order least-squares method

“…Various error estimates are obtained for least-squares methods similar to the one we discuss in this paper; see, e.g. Manteuffel et al [10], and Pehlivanov et al [12].…”

“…This is not the case for [4]. For other least-squares type methods, we refer to [2,7,10,12] and the references therein.…”

Stabilized hybrid finite element methods based on the combination of saddle point principles of elasticity problems

“…The paper shows that this combination leads to a homology family of optimality conditions for the two variational principles, and the new hybrid finite element method does not require the inf-sup condition. Thus the method is stabilized, but different from the various least square methods [3,4,5,8]. For convenience, it will be referred to as the combined hybrid method.…”