Non-Conforming Mesh Refinement for High-Order Finite Elements

We propose a general algorithm for non-conforming adaptive mesh refinement (AMR) of unstructured meshes in high-order finite element codes. Our focus is on h-refinement with a fixed polynomial order. The algorithm handles triangular, quadrilateral, hexahedral and prismatic meshes of arbitrarily high order curvature, for any order finite element space in the de Rham sequence. We present a flexible data structure for meshes with hanging nodes and a general procedure to construct the conforming interpolation operator, both in serial and in parallel. The algorithm and data structure allow anisotropic refinement of tensor product elements in 2D and 3D, and support unlimited refinement ratios of adjacent elements. We report numerical experiments verifying the correctness of the algorithms, and perform a parallel scaling study to show that we can adapt meshes containing billions of elements and run efficiently on 393,000 parallel tasks. Finally, we illustrate the integration of dynamic AMR into a high-order Lagrangian hydrodynamics solver.

[1]  Michael Griebel,et al.  Parallel multigrid in an adaptive PDE solver based on hashing and space-filling curves , 1999, Parallel Comput..

[2]  Ümit V. Çatalyürek,et al.  The Zoltan and Isorropia parallel toolkits for combinatorial scientific computing: Partitioning, ordering and coloring , 2012, Sci. Program..

[3]  Panayot S. Vassilevski,et al.  Algebraic Hybridization and Static Condensation with Application to Scalable H(div) Preconditioning , 2018, SIAM J. Sci. Comput..

[4]  William F. Mitchell,et al.  Hamiltonian Paths Through Two- and Three-Dimensional Grids , 2005, Journal of research of the National Institute of Standards and Technology.

[5]  Carsten Carstensen,et al.  Hanging nodes in the unifying theory of a posteriori finite element error control , 2009 .

[6]  Vipin Kumar,et al.  A Parallel Algorithm for Multilevel Graph Partitioning and Sparse Matrix Ordering , 1998, J. Parallel Distributed Comput..

[7]  Miriam Mehl,et al.  Peano - A Traversal and Storage Scheme for Octree-Like Adaptive Cartesian Multiscale Grids , 2011, SIAM J. Sci. Comput..

[8]  E. Wilson.  The static condensation algorithm , 1974 .

[9]  Boleslaw K. Szymanski,et al.  Adaptive Local Refinement with Octree Load Balancing for the Parallel Solution of Three-Dimensional Conservation Laws , 1997, J. Parallel Distributed Comput..

[10]  Carsten Burstedde,et al.  Recursive Algorithms for Distributed Forests of Octrees , 2014, SIAM J. Sci. Comput..

[11]  William F. Mitchell,et al.  A refinement-tree based partitioning method for dynamic load balancing with adaptively refined grids , 2007, J. Parallel Distributed Comput..

[12]  D. Brandt,et al.  Multi-level adaptive solutions to boundary-value problems math comptr , 1977 .

[13]  Leszek Demkowicz,et al.  Toward a universal h-p adaptive finite element strategy , 1989 .

[14]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[15]  F. T. Suttmeier.  On concepts of PDE-Software: The cellwise oriented approach in DEAL , 2007 .

[16]  W. Bangerth,et al.  deal.II—A general-purpose object-oriented finite element library , 2007, TOMS.

[17]  Robert D. Falgout,et al.  hypre: A Library of High Performance Preconditioners , 2002, International Conference on Computational Science.

[18]  Bram Reps,et al.  Complex Additive Geometric Multilevel Solvers for Helmholtz Equations on Spacetrees , 2015, ACM Trans. Math. Softw..

[19]  Carsten Burstedde,et al.  p4est: Scalable Algorithms for Parallel Adaptive Mesh Refinement on Forests of Octrees , 2011, SIAM J. Sci. Comput..

[20]  Vincent Heuveline,et al.  H1-interpolation on quadrilateral and hexahedral meshes with hanging nodes , 2007, Computing.

[21]  David R. O'Hallaron,et al.  Scalable Parallel Octree Meshing for TeraScale Applications , 2005, ACM/IEEE SC 2005 Conference (SC'05).

[22]  Matthew G. Knepley,et al.  Mesh algorithms for PDE with Sieve I: Mesh distribution , 2009 .

[23]  L Howarth.  Similarity and Dimensional Methods in Mechanics , 1960 .

[24]  Danna Zhou,et al.  d. , 1934, Microbial pathogenesis.

[25]  Ivo Dolezel,et al.  Arbitrary-level hanging nodes and automatic adaptivity in the hp-FEM , 2008, Math. Comput. Simul..

[26]  Lung-an Ying,et al.  Partial differential equations and the finite element method , 2007, Math. Comput..

[27]  P. Solín.  Partial differential equations and the finite element method , 2005 .

[28]  R. Bank,et al.  Some Refinement Algorithms And Data Structures For Regular Local Mesh Refinement , 1983 .

[29]  Benjamin S. Kirk,et al.  Library for Parallel Adaptive Mesh Refinement / Coarsening Simulations , 2006 .

[30]  Matthew G. Knepley,et al.  Support for Non-conformal Meshes in PETSc's DMPlex Interface , 2015, ArXiv.

[31]  Graham F. Carey,et al.  A mesh-refinement scheme for finite element computations , 1976 .

[32]  Marjorie A. McClain,et al.  A Survey of hp-Adaptive Strategies for Elliptic Partial Differential Equations , 2011 .

[33]  P. Campbell,et al.  Dynamic Octree Load Balancing Using Space-Filling Curves ∗ , 2003 .

[34]  Martin Kronbichler,et al.  Algorithms and data structures for massively parallel generic adaptive finite element codes , 2011, ACM Trans. Math. Softw..

[35]  Tzanio V. Kolev,et al.  High-Order Curvilinear Finite Element Methods for Lagrangian Hydrodynamics , 2012, SIAM J. Sci. Comput..

[36]  Ivo Babuška,et al.  Reliable error estimation and mesh adaptation for the finite element method , 1979 .

[37]  Robert Schneiders,et al.  Quadrilateral and Hexahedral Element Meshes , 2002 .

[38]  T. Schönfeld.  ADAPTIVE MESH REFINEMENT METHODS FOR THREE-DIMENSIONAL INVISCID FLOW PROBLEMS , 1995 .

[39]  M. Ainsworth,et al.  Aspects of an adaptive hp-finite element method : Adaptive strategy, conforming approximation and efficient solvers , 1997 .