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A p-Multigrid Strategy with Anisotropic p-Adaptation Based on Truncation Errors for High-Order Discontinuous Galerkin Methods [article]

Andrés M. Rueda-Ramírez, Juan Manzanero, Esteban Ferrer, Gonzalo Rubio, Eusebio Valero
2018 pre-print
In this paper, we combine both methods and present a novel anisotropic p-adaptation multigrid algorithm for steady-state problems that uses the multigrid scheme both as a solver and as an anisotropic error  ...  On the one hand, local mesh adaptation strategies based on error estimation have been proposed to reduce the number of degrees of freedom while keeping a similar accuracy.  ...  Acknowledgments The authors would like to thank David Kopriva for his friendly advise and cooperation.  ... 
doi:10.1016/j.jcp.2018.11.009 arXiv:1806.11456v1 fatcat:z3lnvh3nszhixa2hyogy6xwruy

An Anisotropic p-Adaptation Multigrid Scheme for Discontinuous Galerkin Methods [chapter]

Andrés M. Rueda-Ramírez, Gonzalo Rubio, Esteban Ferrer, Eusebio Valero
2020 Lecture Notes in Computational Science and Engineering  
Acknowledgements This project has received funding from the European Union's Horizon 2020 Research and Innovation Program under the Marie Skłodowska-Curie grant agreement No 675008 for the SSeMID project  ...  Every time the error is estimated, an anisotropic p-multigrid strategy is used to generate a truncation error map for each element, like the one in Fig. 1a .  ...  Introduction In recent decades, high-order discontinuous Galerkin (DG) methods have been gaining increasing popularity for high-accuracy solutions of systems of conservation laws, such as the compressible  ... 
doi:10.1007/978-3-030-39647-3_44 fatcat:wll2gknan5ahhkqzr3rs4gl3bi

Numerical Stabilization of the Melt Front for Laser Beam Cutting [chapter]

Torsten Adolph, Willi Schönauer, Markus Niessen, Wolfgang Schulz
2010 Numerical Mathematics and Advanced Applications 2009  
We develop a hybrid scheme consisting of a combination of a second order MUSCL scheme and a high order finite difference scheme.  ...  The model is based on Navier-Stokes equations for viscous incompressible flow and convection-diffusion equation describing pollution transport.  ...  problem, based on (P k 2 , P k-1 ) discontinuous elements for k = 1, 2 or 3.  ... 
doi:10.1007/978-3-642-11795-4_6 fatcat:nx4nvuxaxfbcdjknopny53ck5e

p-Multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier–Stokes equations

Krzysztof J. Fidkowski, Todd A. Oliver, James Lu, David L. Darmofal
2005 Journal of Computational Physics  
We present a p-multigrid solution algorithm for a high-order discontinuous Galerkin finite element discretization of the compressible Navier-Stokes equations.  ...  Fourier analysis of the two-level p-multigrid algorithm for convection-diffusion shows that element line Jacobi presents a significant improvement over element Jacobi especially for high Reynolds number  ...  A portion of this work was supported by NASA contract NAG-1-02037.  ... 
doi:10.1016/j.jcp.2005.01.005 fatcat:tsjnuxsa5bgd7e2fmilsctffd4

Smoothed Aggregation Multigrid for the Discontinuous Galerkin Method

F. Prill, M. Lukáčová-Medviďová, R. Hartmann
2009 SIAM Journal on Scientific Computing  
The discontinuous Galerkin method (dGFEM) has become popular for the discretization of elliptic and hyperbolic partial differential equations for about a decade.  ...  The method is scalable, if also the computational work for a problem containing n unknowns grows like O(n log n). Multigrid methods are known to be scalable for elliptic problems on regular grids.  ...  The multigrid algorithm needs to be combined with a p-multilevel method or some related domain decomposition approach to yield a hierarchical solver for high order numerical systems.  ... 
doi:10.1137/080728457 fatcat:akcpit7x4vcbtos4ucgixfee2y

UNSTRUCTURED GRID TECHNIQUES

D. J. Mavriplis
1997 Annual Review of Fluid Mechanics  
The topics of mesh generation and adaptation, spatial discretization, and solution techniques for steady flows are covered.  ...  Remaining difficulties in these areas are highlighted, and directions for future work are outlined.  ...  strategies with multigrid methods.  ... 
doi:10.1146/annurev.fluid.29.1.473 fatcat:n4c2wf6prbeyvnscis762jugtm

A review of algebraic multigrid

K. Stüben
2001 Journal of Computational and Applied Mathematics  
This paper gives a review of the ÿrst hierarchical and purely matrix-based approach, algebraic multigrid (AMG).  ...  For practically relevant problem sizes, classical one-level methods had already reached their limits and new hierarchical algorithms had to be developed in order to allow an e cient solution of even larger  ...  In practice, we want to achieve rapid convergence with as small sets of interpolatory variables P i as possible (in order to allow for a rapid coarsening and to obtain reasonably sparse Galerkin operators  ... 
doi:10.1016/s0377-0427(00)00516-1 fatcat:q2kwtxvbmbakpgnbxihbplrece

Delayed approximate matrix assembly in multigrid with dynamic precisions [article]

Charles D. Murray, Tobias Weinzierl
2020 arXiv   pre-print
matrix entries carry little information or are not yet available with high accuracy.  ...  The accurate assembly of the system matrix is an important step in any code that solves partial differential equations on a mesh.  ...  We use our second test setup with only one type of a discontinuous material jump over three orders of magnitude. Initial results are given for ∈ {1, 10 −3 } (Fig. 10 ).  ... 
arXiv:2005.03606v1 fatcat:iyad6selxfeeflgp7xdq7tawum

Scalable angular adaptivity for Boltzmann transport [article]

S. Dargaville, A.G. Buchan, R.P. Smedley-Stevenson, P.N Smith, C.C. Pain
2019 arXiv   pre-print
This adaptivity uses Haar wavelets, which perform structured h-adaptivity built on top of a hierarchical P_0 FEM discretisation of a 2D angular domain, allowing different anisotropic angular resolution  ...  Fixed angular refinement, along with regular and goal-based error metrics are shown in three example problems taken from neutronics/radiative transfer applications.  ...  This highly anisotropic problem shows the need for low-order adaptive methods, as we can see in Fig. 20a that the convergence of a spectral P n method has degraded to roughly that of the uniform P 0  ... 
arXiv:1901.04929v1 fatcat:rkpzwy4dtvbmddjbls6p5k2sgi

Error Estimation and Adaptive Mesh Refinement for Aerodynamic Flows [chapter]

Ralf Hartmann, Joachim Held, Tobias Leicht, Florian Prill
2010 Notes on Numerical Fluid Mechanics and Multidisciplinary Design  
One extremely promising class of high-order schemes based on the finite element frame- work are Discontinuous Galerkin (DG, for short) methods.  ...  A. Oliver, J. Lu, and D. L. Darmofal. p-multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations. J. Comp.  ... 
doi:10.1007/978-3-642-03707-8_24 fatcat:2m2ao7uwivaeropjt55kdht43u

Filtering algebraic multigrid and adaptive strategies

Arne Nägel, Robert D. Falgout, Gabriel Wittum
2007 Computing and Visualization in Science  
This paper proposes a variant of Wagner's method with substantially improved robustness properties.  ...  Due to shortcomings of geometric approaches, algebraic multigrid methods have been developed. One example is the filtering algebraic multigrid method introduced by C. Wagner.  ...  Based on [8] we propose an adaptive strategy using the latest correction as test vector when the convergence rate stalls.  ... 
doi:10.1007/s00791-007-0066-9 fatcat:s7gftzdxgvbldlqead3a6kugam

A computational study of turbulent flow separation for a circular cylinder using skin friction boundary conditions [chapter]

Johan Hoffman, Niclas Jansson
2011 ERCOFTAC Series  
A priori estimates are applied to the anisotropic mesh adaptation for 2D viscous flows and 3D ones including Large Eddy Simulation.  ...  The BE poses a formidable challenge for numerical approximation methods, on account of its high dimensional phase-space setting: for a problem in N spatial dimensions, the single molecule phase-space is  ...  made possible the publication of this book of Abstracts, the (online or CD) publication of the Extended Abstracts and Full Contributions, and the organization of ADMOS 2013, VI International Conference on  ... 
doi:10.1007/978-94-007-0231-8_5 fatcat:f6ll6iawcbeo7pbj4rb7t4o2tq

Finite Element Methods for the Incompressible Navier-Stokes Equations [chapter]

Rolf Rannacher
2000 Fundamental Directions in Mathematical Fluid Mechanics  
These notes are based on lectures given in a Short Course on Theoretical and Numerical  ...  Design such a method for high-order schemes.  ...  (I) The traditional method for a posteriori time-step selection is based on the concept of controlling the local "truncation error" but neglecting the global error accumulation.  ... 
doi:10.1007/978-3-0348-8424-2_6 fatcat:vf5hwpwjyfe6hjwf5hs6772wf4

A Matrix Dependent/Algebraic Multigrid Approach for Extruded Meshes with Applications to Ice Sheet Modeling

R. Tuminaro, M. Perego, I. Tezaur, A. Salinger, S. Price
2016 SIAM Journal on Scientific Computing  
based on the first-order Stokes equations [46] .  ...  A multigrid method is proposed that combines ideas from matrix dependent multigrid for structured grids and algebraic multigrid for unstructured grids.  ...  We thank Ali Pinar for suggesting the idea of combining coloring and connected component algorithms to detect hinged peninsulas within graphs.  ... 
doi:10.1137/15m1040839 fatcat:pvpdnaoh2rcoldrob5g4pbdr4m

Quasi-matrix-free Hybrid Multigrid on Dynamically Adaptive Cartesian Grids

Marion Weinzierl, Tobias Weinzierl
2018 ACM Transactions on Mathematical Software  
They align with matrix-free geometric multigrid solvers as they never assemble the system matrices which is cumbersome for dynamically adaptive grids and full multigrid.  ...  We present a family of spacetree-based multigrid realizations using the tree's multiscale nature to derive coarse grids.  ...  Marion wants to thank him for hosting her research stay at Technion which laid the foundations for the present work.  ... 
doi:10.1145/3165280 fatcat:wmygp64iy5f3bbnwiu5l4g6bfq
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