The Surrogate Matrix Methodology: A Priori Error Estimation

We give the first mathematically rigorous analysis of an emerging approach to finite element analysis (see, e.g., Bauer et al. [Appl. Numer. Math., 122 (2017), pp. 14--38]), which we hereby refer t...

[1]  John Loffeld,et al.  On the arithmetic intensity of high-order finite-volume discretizations for hyperbolic systems of conservation laws , 2019, Int. J. High Perform. Comput. Appl..

[2]  Dave A. May,et al.  A scalable, matrix-free multigrid preconditioner for finite element discretizations of heterogeneous Stokes flow , 2015 .

[3]  Barbara I. Wohlmuth,et al.  Large-scale simulation of mantle convection based on a new matrix-free approach , 2019, J. Comput. Sci..

[4]  Ulrich Rüde,et al.  The HyTeG finite-element software framework for scalable multigrid solvers , 2018, Int. J. Parallel Emergent Distributed Syst..

[5]  Katharina Kormann,et al.  A generic interface for parallel cell-based finite element operator application , 2012 .

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

[7]  Ulrich Rüde,et al.  Towards Textbook Efficiency for Parallel Multigrid , 2015 .

[8]  Ralph Müller,et al.  A scalable multi‐level preconditioner for matrix‐free µ‐finite element analysis of human bone structures , 2008 .

[9]  J. Guermond,et al.  Theory and practice of finite elements , 2004 .

[10]  Cyril Flaig,et al.  A Highly Scalable Matrix-Free Multigrid Solver for μFE Analysis Based on a Pointer-Less Octree , 2011, LSSC.

[11]  Barbara I. Wohlmuth,et al.  A two-scale approach for efficient on-the-fly operator assembly in massively parallel high performance multigrid codes , 2016, ArXiv.

[12]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[13]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[14]  Martin Kronbichler,et al.  Multigrid for matrix-free finite element computations on graphics processors , 2017 .

[15]  Jed Brown,et al.  Efficient Nonlinear Solvers for Nodal High-Order Finite Elements in 3D , 2010, J. Sci. Comput..

[16]  O. Ghattas,et al.  Parallel Octree-Based Finite Element Method for Large-Scale Earthquake Ground Motion Simulation , 2005 .

[17]  G. Carey,et al.  Element‐by‐element linear and nonlinear solution schemes , 1986 .

[18]  Benjamin Karl Bergen,et al.  Hierarchical hybrid grids: data structures and core algorithms for multigrid , 2004, Numer. Linear Algebra Appl..

[19]  Karl Ljungkvist,et al.  Matrix-free finite-element computations on graphics processors with adaptively refined unstructured meshes , 2017, SpringSim.

[20]  Matthew G. Knepley,et al.  Extreme-Scale Multigrid Components within PETSc , 2016, PASC.

[21]  Barbara I. Wohlmuth,et al.  A New Matrix-Free Approach for Large-Scale Geodynamic Simulations and its Performance , 2018, ICCS.

[22]  A. R. Mitchell,et al.  Curved elements in the finite element method , 1974 .

[23]  Barbara I. Wohlmuth,et al.  A stencil scaling approach for accelerating matrix-free finite element implementations , 2017, SIAM J. Sci. Comput..

[24]  P. G. Ciarlet,et al.  Three-dimensional elasticity , 1988 .

[25]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .

[26]  William Gropp,et al.  Efficient Management of Parallelism in Object-Oriented Numerical Software Libraries , 1997, SciTools.

[27]  Robert D. Falgout,et al.  Stencil computations for PDE‐based applications with examples from DUNE and hypre , 2017, Concurr. Comput. Pract. Exp..

[28]  Patrick Amestoy,et al.  A Fully Asynchronous Multifrontal Solver Using Distributed Dynamic Scheduling , 2001, SIAM J. Matrix Anal. Appl..

[29]  Patrick Le Tallec,et al.  Coupled variational formulations of linear elasticity and the DPG methodology , 2016, J. Comput. Phys..

[30]  Benjamin Karl Bergen,et al.  Hierarchical hybrid grids: data structures and core algorithms for efficient finite element simulations on supercomputers = Hierarchische hybride Gitter , 2005 .

[31]  Patrick Amestoy,et al.  Hybrid scheduling for the parallel solution of linear systems , 2006, Parallel Comput..

[32]  J. Miller.  Numerical Analysis , 1966, Nature.

[33]  J. Bey,et al.  Tetrahedral grid refinement , 1995, Computing.

[34]  Ulrich Rüde,et al.  Hierarchical hybrid grids: achieving TERAFLOP performance on large scale finite element simulations , 2007, Int. J. Parallel Emergent Distributed Syst..

[35]  B. van Rietbergen,et al.  COMPUTATIONAL STRATEGIES FOR ITERATIVE SOLUTIONS OF LARGE FEM APPLICATIONS EMPLOYING VOXEL DATA , 1996 .

[36]  L. R. Scott,et al.  Finite element interpolation of nonsmooth functions satisfying boundary conditions , 1990 .

[37]  G. Strang.  VARIATIONAL CRIMES IN THE FINITE ELEMENT METHOD , 1972 .

[38]  John W. Barrett,et al.  Finite element approximation of the p-Laplacian , 1993 .