Very High Order Anisotropic Metric-Based Mesh Adaptation in 3D

Abstract

In this paper, we study the extension of anisotropic metric-based mesh adaptation to the case of very high-order solutions in 3D. This work is based on an extension of the continuous mesh framework and multi-scale 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 point-wise leading term of the local error is a 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 ▪. From a geometric point of view, this problem is equivalent to finding a maximal volume ellipsoid included in the level set one of the absolute value of the polynomial. This optimization problem is strongly non-linear both for the functional and the constraints. We first recast the continuous problem in a discrete setting in the metric-logarithm 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 log-simplex problems. From the field of the local quadratic forms that representing the high-order 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.