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Optimally adapted finite elements meshes

Given a function f deﬁned on a bounded domain Ω ⊂ IR 2 and a number N > 0, we study the properties of the triangulation T N that minimizes the distance between f and its interpolation on the associated ﬁnite element space, over all triangulations of at most N elements. The error is studied in the norm X = L p for 1 ≤ p ≤ ∞ and we consider Lagrange ﬁnite elements of arbitrary polynomial degree m − 1. We establish sharp asymptotic error estimates as N → + ∞ when the optimal anisotropic triangulation is used, recovering the results on piecewise linear interpolation [3, 4, 12], an improving the results on higher degree interpolation [9, 10, 11]. These estimates involve invariant polynomials applied to the m -th order derivatives of f . In addition, our analysis also provides with practical strategies for designing meshes such that the interpolation error satisﬁes the optimal estimate up to a ﬁxed multiplicative constant. We partially extend our results to higher dimensions for ﬁnite elements on simplicial partitions of a domain Ω ⊂ IR d .