Finite Element Approximation of Some Degenerate Monotone Quasilinear Elliptic Systems

In this paper we examine the continuous piecewise linear finite element approximation of the following system: given f ≡ (fⱼ ) and g ≡ (gⱼ ), find u ≡ (uⱼ )((j = 1 → r with $r = 1$ or 2) such that - ∇ · (K(z,∇ u(z))∇ u(z)) = f(z), z ∈ Ω ⊂ R² , u |∂ Ω = g |∂ Ω , where (∇ u) ≡ ∂ uⱼ / ∂ zᵢ \, 1 ≤ i ≤ 2,\, 1 ≤ j ≤ r and K is a given matrix on Ω × R2 × r. We characterize a class of matrices K for which we prove error bounds for this discretization. For sufficiently regular solutions u, achievable at least for some model problems, our bounds improve on existing results in the literature. It is shown that for a notable subclass of K, for which only suboptimal error bounds have been previously derived, the piecewise linear finite element approximation of this problem will converge at the optimal rate in an energy-type norm. It is also shown that the techniques used in this paper can be applied to more general problems.