Surface mesh optimization, adaption, and untangling with high-order accuracy

We investigate the problem of optimizing, adapting, and untangling a surface triangulation with high-order accuracy, so that the resulting mesh has sufficient accuracy for high-order numerical methods, such as finite element methods with quadratic or cubic elements or generalized finite difference methods. We show that low-order remeshing, which may preserve the "shape" of the surface, can undermine the order of accuracy or even cause non-convergence of numerical computations. In addition, most existing methods are incapable of accurately remeshing surface meshes with inverted elements. We describe a remeshing strategy that can produce high-quality triangular meshes, while untangling mildly folded triangles and preserving the geometry to high-order accuracy. Our approach extends our earlier work on high-order surface reconstruction and mesh optimization. We present the theoretical framework of our methods, experimental comparisons against other methods, and demonstrate its utilization in accurate solutions for geometric partial differential equations on triangulated surfaces.