Error estimates for Raviart-Thomas interpolation of any order on anisotropic tetrahedra

GABRIEL ACOSTA; THOMAS APEL; RICARDO G. DUR¨¢N; ARIEL L. LOMBARDI

MATHEMATICS OF COMPUTATION

AMER MATHEMATICAL SOC

Lugar: Providence; Año: 2011 vol. 80 p. 141 - 163

0025-5718

We prove optimal order error estimates for the Raviart-Thomasinterpolation of arbitrary order under the maximum angle condition for trianglesand under two generalizations of this condition, namely, the so-calledthree-dimensional maximum angle condition and the regular vertex property,for tetrahedra.Our techniques are different from those used in previous papers on thesubject, and the results obtained are more general in several aspects. First,intermediate regularity is allowed; that is, for the Raviart-Thomas interpolationof degree k ¡Ý 0, we prove error estimates of order j + 1 when the vectorfield being approximated has components in W^{j+1,p}, for triangles or tetrahedra,where 0 ¡Ü j ¡Ü k and 1 ¡Ü p ¡Ü ¡Þ. These results are new even in thetwo-dimensional case. Indeed, the estimate was known only in the case j = k.On the other hand, in the three-dimensional case, results under the maximumangle condition were known only for k = 0.