Piece-wise constant approximations in variational problems via W^1,p estimates

Piece-wise constant approximation of first order variational problems via W −1, p estimates C. Davini, R. Paroni Dipartimento di Georisorse e Territorio, Via Cotonificio 114, 33100 Udine, Italy DAP, Università di Sassari, Palazzo del Pou Salit, 07041 Alghero, Italy A classical problem in the calculus of variations is: find the minimizers of the functional F(v) = W (x, v, ∇v) dx – (f , v) Ω among all functions v ∈ W 1, p(Ω) with trace equal to w ∈ W 1, p(Ω) over a subset (of positive length) ∂uΩ of the boundary of Ω, where 1 < p < +∞, Ω ⊂ R2 is an open bounded set with Lipschitz boundary, f ∈ Lq(Ω) and W : Ω × R × R2 → R is a Carathéodory function convex in the last variable and satisfying a standard p-growth from below and above. Different schemes have been developed in order to find an approximation of the minimizer(s) of the problem above. Probably, the most popular is the technique based on the use of continuous piece-wise affine finite elements. Higher order approximants have also been used. These on one hand give a better rate of convergence but on the other hand make the numerical scheme more complex. Our point of view here is to consider the space which makes the numerical scheme as simple as possible, which is the space of piece-wise constant functions over triangulations of the base domain. By using piece-wise constant functions the first problem at our hand is to define what we mean by gradient. At each nodal point xi of the triangulation of the base domain we call generalized gradient of a piece-wise constant function a suitable mean of the distributional gradient on a dual element around the point xi. This notion is then extended to the full triangulation by taking the generalized gradient constant on each dual element. Despite the given name, the generalized gradient is not a gradient, even though we show that it has some of the properties which are peculiar to a gradient. In particular we prove that if a sequence of generalized gradients weakly converges in Lp then the weak limit is a gradient. The generalized gradient instead does not have the "imbedding property" of a gradient, which is: a sequence of piece-wise constant functions which weakly converges together with the sequence of the generalized gradients does not necessarily strongly converges. This property is recovered by requiring that a certain weighted Lp norm of the jumps of the piece-wise constant function across the edges of the mesh should tend to zero as the size of the triangulation goes to zero. This is proved by working in the W −1, p space and using an inequality due to Necˇas. The lacking of this imbedding property strongly influences the definition of the discrete functionals which approximate the original one. References [1] Davini, C.; Paroni, R. External approximation of first order variational problems via W −1, p estimates. ESAIM Control Optim. Calc. Var. 14 (2008), no. 4, 802–824.