A generic interface for parallel and adaptive discretization schemes: abstraction principles and the Dune-Fem module

Starting from an abstract mathematical notion of discrete function spaces and operators, we derive a general abstraction for a large class of grid-based discretization schemes for stationary and instationary partial differential equations. Special emphasis is put on concepts for local adaptivity and parallelization with dynamic load balancing. The concepts are based on a corresponding abstract definition of a parallel and hierarchical adaptive grid given in Bastian et al. (Computing 82(2–3):103–119, 2008). Based on the abstract framework, we describe an efficient object oriented implementation of a generic interface for grid-based discretization schemes that is realized in the Dune-Fem library (http://dune.mathematik.uni-freiburg.de). By using interface classes we manage to separate functionality from data structures. Efficiency is obtained by using modern template based generic programming techniques, including static polymorphism, the engine concept, and template metaprogramming. We present numerical results for several benchmark problems and some advanced applications.

[1]  D. Kröner.  Numerical Schemes for Conservation Laws , 1997 .

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

[3]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[4]  Michael Christian Lehn,et al.  FLENS - a flexible library for efficient numerical solutions , 2008 .

[5]  J. Hesthaven,et al.  Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations , 2007 .

[6]  Patrick Henning,et al.  The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift , 2010, Networks Heterog. Media.

[7]  Andreas Dedner,et al.  A generic grid interface for parallel and adaptive scientific computing. Part I: abstract framework , 2008, Computing.

[8]  R. Eymard,et al.  Finite Volume Methods , 2019, Computational Methods for Fluid Dynamics.

[9]  George Karypis,et al.  Multilevel k-way Partitioning Scheme for Irregular Graphs , 1998, J. Parallel Distributed Comput..

[10]  C. Rohde,et al.  A parallel, load-balanced MHD code on locally-adapted, unstructured grids in 3d , 2004 .

[11]  Jean-Frédéric Gerbeau,et al.  Derivation of viscous Saint-Venant system for laminar shallow water , 2001 .

[12]  Andreas Dedner,et al.  A generic grid interface for parallel and adaptive scientific computing. Part II: implementation and tests in DUNE , 2008, Computing.

[13]  Andreas Dedner,et al.  A Generic Stabilization Approach for Higher Order Discontinuous Galerkin Methods for Convection Dominated Problems , 2011, J. Sci. Comput..

[14]  Daniel A. White,et al.  FEMSTER: An object-oriented class library of high-order discrete differential forms , 2005, TOMS.

[15]  Chi-Wang Shu,et al.  Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems , 2001, J. Sci. Comput..

[16]  Andreas Dedner,et al.  A general object oriented framework for discretizing non-linear evolution equations , 2006 .

[17]  Peter Bastian,et al.  The Iterative Solver Template Library , 2006, PARA.

[18]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[19]  Vipin Kumar,et al.  Parallel Multilevel series k-Way Partitioning Scheme for Irregular Graphs , 1999, SIAM Rev..

[20]  Timothy A. Davis,et al.  Algorithm 832: UMFPACK V4.3---an unsymmetric-pattern multifrontal method , 2004, TOMS.

[21]  Andreas Dedner,et al.  Curvature Guided Level Set Registration Using Adaptive Finite Elements , 2007, DAGM-Symposium.

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

[23]  Jack Dongarra,et al.  Applied Parallel Computing. State of the Art in Scientific Computing, 8th International Workshop, PARA 2006, Umeå, Sweden, June 18-21, 2006, Revised Selected Papers , 2007, PARA.

[24]  Yazid Abdelaziz,et al.  Review: A survey of the extended finite element , 2008 .

[25]  Andreas Dedner,et al.  An efficient implementation of an adaptive and parallel grid in DUNE , 2006 .

[26]  Vipin Kumar,et al.  A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs , 1998, SIAM J. Sci. Comput..

[27]  Kunibert G. Siebert,et al.  Design of Adaptive Finite Element Software - The Finite Element Toolbox ALBERTA , 2005, Lecture Notes in Computational Science and Engineering.

[28]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[29]  H. Rentz-Reichert,et al.  UG – A flexible software toolbox for solving partial differential equations , 1997 .

[30]  B. Haasdonk,et al.  REDUCED BASIS METHOD FOR FINITE VOLUME APPROXIMATIONS OF PARAMETRIZED LINEAR EVOLUTION EQUATIONS , 2008 .

[31]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[32]  P. Henning,et al.  A-posteriori error estimate for a heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift , 2009 .