
Very High Order Anisotropic MetricBased Mesh Adaptation in 3D
Coulaud, Olivier, Adrien Loseille
Proceedings, 25th International Meshing Roundtable, Elsevier, Science Direct, September 2630 2016

INTERNATIONAL MESHING ROUNTABLE

25th International Meshing Roundtable
Washington DC, U.S.A.
September 2630, 2016
Olivier Coulaud, INRIA, FR, olivier.a.coulaud@inria.fr
Adrien Loseille, INRIA, FR, adrien.loseille@inria.fr
Abstract
In this paper, we study the extension of anisotropic metricbased mesh adaptation to the case of very highorder solutions in 3D.
This work is based on an extension of the continuous mesh framework and multiscale mesh adaptation [10] where the optimal
metric is derived through a calculus of variation. Based on classical high order a priori error estimates [4], the pointwise leading
term of the local error is an homogeneous polynomial of order k + 1. To derive the leading anisotropic direction and orientations,
this polynomial is approximated by a quadratic positive definite form, taken to the power k+1 . From a geometric point of view, this 2
problem is equivalent to found a maximal volume ellipsoid included in the level set one of the absolute value of the polynomial. This optimization problem is strongly nonlinear both for the functional and the constraints. We first recast the continuous problem in a discrete setting in the metriclogarithm space. With this approximation, this problem becomes linear and is solved with the simplex algorithm [5]. This optimal quadratic form in the Euclidean space is then found by iteratively solving a sequence of such logsimplex problems. From the field of the local quadratic forms that representing the highorder error, a calculus of variation is used to globally control the error in Lp norm. A closed form of the optimal metric is then found. Anisotropic meshes are then generated with this metric based on the unit mesh concept [8]. For the numerical experiments, we consider several analytical functions in 3D. Convergence rate and optimality of the meshes are then discussed for interpolation of orders 1 to 5.
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