%
% This file was created by the Typo3 extension
% sevenpack version 0.7.10
%
% --- Timezone: UTC
% Creation date: 2019-08-23
% Creation time: 17-28-32
% --- Number of references
% 63
%
@Article { MR3855138,
title = {Orbital-enriched flat-top partition of unity method for the Schr{\~A}¶dinger eigenproblem},
journal = {Comput. Methods Appl. Mech. Engrg.},
year = {2018},
volume = {342},
pages = {224--239},
DOI = {10.1016/j.cma.2018.07.042},
author = {Albrecht, Clelia and Klaar, Constanze and Pask, John Ernest and Schweitzer, Marc Alexander and Sukumar, N. and Ziegenhagel, Albert}
}
@Article { BULER201745,
title = {Visualization of fracture progression in peridynamics},
journal = {Computers \& Graphics},
year = {2017},
volume = {67},
number = {Supplement C},
pages = {45 - 57},
ISSN = {0097-8493},
DOI = {https://doi.org/10.1016/j.cag.2017.05.003},
author = {Bussler, Michael and Diehl, Patrick and Pfl{\~A}¼ger, Dirk and Frey, Steffen and Sadlo, Filip and Ertl, Thomas and Alexander Schweitzer, Marc}
}
@Book { Griebel.Schueller.Schweitzer:2017,
title = {Scientific Computing and Algorithms in Industrial Simulations:Projects and Products of Fraunhofer SCAI},
year = {2017},
editor = {M. Griebel and A. Sch{\~A}¼ler and M.\verb=~=A. Schweitzer},
publisher = {Springer}
}
@Inbook { Plum:Krechel:etal:2017,
title = {Scientific Computing and Algorithms in Industrial Simulations: Projects and Products of Fraunhofer SCAI},
year = {2017},
pages = {121--134},
editor = {M. Griebel and A. Schueller and M. A. Schweitzer},
publisher = {Springer International Publishing},
chapter = {Parallel Algebraic Multigrid},
author = {Plum, Hans-Joachim and Krechel, Arnold and Gries, Sebastian and Metsch, Bram and Nick, Fabian and Alexander Schweitzer, Marc and St{\~A}¼ben, Klaus}
}
@Inbook { Schweitzer:Ziegenhagel:2017,
title = {Scientific Computing and Algorithms in Industrial Simulations: Projects and Products of Fraunhofer SCAI},
year = {2017},
pages = {207--228},
editor = {M. Griebel and A. Sch{\~A}¼ller and M. A. Schweitzer},
publisher = {Springer International Publishing},
chapter = {Rapid Enriched Simulation Application Development with PUMA},
author = {A. Schweitzer, M. and Ziegenhagel, A.}
}
@Incollection { MR3718334,
title = {Embedding enriched partition of unity approximations in finite element simulations},
year = {2017},
volume = {115},
pages = {199--208},
publisher = {Springer, Cham},
series = {Lect. Notes Comput. Sci. Eng.},
booktitle = {Meshfree methods for partial differential equations VIII},
author = {Schweitzer, Marc Alexander and Ziegenhagel, Albert}
}
@Incollection { MR3718333,
title = {Evaluation of local multiscale approximation spaces for partition of unity methods},
year = {2017},
volume = {115},
pages = {167--198},
publisher = {Springer, Cham},
series = {Lect. Notes Comput. Sci. Eng.},
booktitle = {Meshfree methods for partial differential equations VIII},
author = {Schweitzer, Marc Alexander and Wu, Sa}
}
@Incollection { MR3718326,
title = {Extraction of fragments and waves after impact damage in particle-based simulations},
year = {2017},
volume = {115},
pages = {17--34},
publisher = {Springer, Cham},
series = {Lect. Notes Comput. Sci. Eng.},
booktitle = {Meshfree methods for partial differential equations VIII},
author = {Diehl, Patrick and Bu{\~A}? ler, Michael and Pfl{\~A}¼ger, Dirk and Frey, Steffen and Ertl, Thomas and Sadlo, Filip and Schweitzer, Marc Alexander}
}
@Incollection { MR3728544,
title = {Parallel algebraic multigrid},
year = {2017},
pages = {121--151},
publisher = {Springer, Cham},
booktitle = {Scientific computing and algorithms in industrial simulations},
author = {Plum, Hans-Joachim and Krechel, Arnold and Gries, Sebastian and Metsch, Bram and Nick, Fabian and Schweitzer, Marc Alexander and St{\~A}¼ben, Klaus}
}
@Incollection { MR3728547,
title = {Rapid enriched simulation application development with PUMA},
year = {2017},
pages = {207--226},
publisher = {Springer, Cham},
booktitle = {Scientific computing and algorithms in industrial simulations},
author = {Schweitzer, Marc Alexander and Ziegenhagel, Albert}
}
@Proceedings { Griebel.Schweitzer:2017,
title = {Meshfree Methods for Partial Differential Equations VIII},
year = {2017},
volume = {115},
editor = {M. Griebel and M.\verb=~=A. Schweitzer},
publisher = {Springer},
series = {Lecture Notes in Computational Science and Engineering}
}
@Inproceedings { Schweitzer.Ziegenhagel:2017,
title = {Embedding Enriched Partition of Unity Approximations inFinite Element Simulations},
year = {2017},
volume = {115},
pages = {195--204},
editor = {M.\verb=~=Griebel and M.\verb=~=A.\verb=~=Schweitzer},
publisher = {Springer International Publishing},
series = {Lecture Notes in Science and Engineering},
booktitle = {Meshfree Methods for Partial Differential Equations VIII},
author = {M.\verb=~=A.\verb=~=Schweitzer and A.\verb=~=Ziegenhagel}
}
@Inproceedings { Schweitzer.Wu:2017,
title = {Evaluation of local multiscale approximation spaces for partition of unity methods},
year = {2017},
volume = {115},
pages = {163--194},
editor = {M.\verb=~=Griebel and M.\verb=~=A.\verb=~=Schweitzer},
publisher = {Springer International Publishing},
series = {Lecture Notes in Science and Engineering},
booktitle = {Meshfree Methods for Partial Differential Equations VIII},
author = {M.\verb=~=A.\verb=~=Schweitzer and S.\verb=~=Wu}
}
@Inproceedings { Heller.Kaiser.Diehl.ea:2016,
title = {Closing the Performance Gap with Modern C++},
year = {2016},
volume = {9945},
pages = {18--31},
editor = {Michaela Taufer and Bernd Mohr and Julian M. Kunkel},
publisher = {Springer International Publishing},
series = {Lecture Notes in Computer Science},
booktitle = {High Performance Computing},
author = {Heller, Thomas and Kaiser, Hartmut and Diehl, Patrick and Fey, Dietmar and Alexander Schweitzer, Marc}
}
@Incollection { MR3587387,
title = {Dispersion properties of the partition of unity method and explicit dynamics},
year = {2015},
volume = {100},
pages = {269--292},
publisher = {Springer, Cham},
series = {Lect. Notes Comput. Sci. Eng.},
booktitle = {Meshfree methods for partial differential equations VII},
author = {Schweitzer, Marc Alexander and Ziegenhagel, Albert}
}
@Incollection { MR3587378,
title = {Efficient neighbor search for particle methods on GPUs},
year = {2015},
volume = {100},
pages = {81--95},
publisher = {Springer, Cham},
series = {Lect. Notes Comput. Sci. Eng.},
booktitle = {Meshfree methods for partial differential equations VII},
author = {Diehl, Patrick and Schweitzer, Marc Alexander}
}
@Incollection { MR3587386,
title = {Numerical integration of on-the-fly-computed enrichment functions in the PUM},
year = {2015},
volume = {100},
pages = {247--267},
publisher = {Springer, Cham},
series = {Lect. Notes Comput. Sci. Eng.},
booktitle = {Meshfree methods for partial differential equations VII},
author = {Schweitzer, Marc Alexander and Wu, Sa}
}
@Inproceedings { Schweitzer.Ziegenhagel:2014,
title = {Dispersion Properties of the Partition of Unity Method \& Explicit Dynamics},
year = {2015},
volume = {100},
note = {Also available as INS Preprint No. 1407},
editor = {M.\verb=~=Griebel and M.\verb=~=A.\verb=~=Schweitzer},
publisher = {Springer},
series = {Lecture Notes in Computational Science and Engineering},
booktitle = {Meshfree Methods for Partial Differential Equations VII},
author = {M.\verb=~=A.\verb=~=Schweitzer and A.\verb=~=Ziegenhagel}
}
@Inproceedings { Diehl.Schweitzer:2014,
title = {Efficient Neighbor Search for Particle Methods on GPUs},
year = {2015},
volume = {100},
note = {Also available as INS Preprint No. 1405},
editor = {M.\verb=~=Griebel and M.\verb=~=A.\verb=~=Schweitzer},
publisher = {Springer},
series = {Lecture Notes in Computational Science and Engineering},
booktitle = {Meshfree Methods for Partial Differential Equations VII},
author = {P.\verb=~=Diehl and M.\verb=~=A.\verb=~=Schweitzer}
}
@Inproceedings { Schweitzer.Wu:2014*1,
title = {Numerical Integration of on-the-fly-computed Enrichment Functions in the PUM},
year = {2015},
volume = {100},
note = {Also available as INS Preprint No. 1406},
editor = {M.\verb=~=Griebel and M.\verb=~=A.\verb=~=Schweitzer},
publisher = {Springer},
series = {Lecture Notes in Computational Science and Engineering},
booktitle = {Meshfree Methods for Partial Differential Equations VII},
author = {M.\verb=~=A.\verb=~=Schweitzer and S.\verb=~=Wu}
}
@Incollection { MR3205048,
title = {A moving least squares approach to the construction of discontinuous enrichment functions},
year = {2014},
pages = {347--360},
publisher = {Springer, Cham},
booktitle = {Singular phenomena and scaling in mathematical models},
DOI = {10.1007/978-3-319-00786-1_15},
author = {Schweitzer, Marc Alexander and Wu, Sa}
}
@Article { MR3042674,
title = {Variational mass lumping in the partition of unity method},
journal = {SIAM J. Sci. Comput.},
year = {2013},
volume = {35},
number = {2},
pages = {A1073--A1097},
DOI = {10.1137/120895561},
author = {Schweitzer, Marc Alexander}
}
@Article { Schweitzer:2013,
title = {Variational Mass Lumping in the Partition of Unity Method},
journal = {SIAM Journal on Scientific Computing},
year = {2013},
volume = {35},
number = {2},
pages = {A1073-A1097},
DOI = {10.1137/120895561},
author = {Schweitzer, M.}
}
@Incollection { MR3288463,
title = {Multilevel partition of unity method for elliptic problems with strongly discontinuous coefficients},
year = {2013},
volume = {89},
pages = {93--109},
publisher = {Springer, Heidelberg},
series = {Lect. Notes Comput. Sci. Eng.},
booktitle = {Meshfree methods for partial differential equations VI},
DOI = {10.1007/978-3-642-32979-1_6},
author = {Schweitzer, Marc Alexander}
}
@Inproceedings { Schweitzer:2013*1,
title = {Multilevel Partition of Unity Method for Elliptic Problems with Strongly Discontinuous Coefficients},
year = {2013},
volume = {89},
pages = {93--110},
editor = {M.\verb=~=Griebel and M.\verb=~=A.\verb=~=Schweitzer},
publisher = {Springer},
series = {Lecture Notes in Computational Science and Engineering},
booktitle = {Meshfree Methods for Partial Differential Equations VI},
author = {M.\verb=~=A.\verb=~=Schweitzer}
}
@Article { MR2863778,
title = {Generalizations of the finite element method},
journal = {Cent. Eur. J. Math.},
year = {2012},
volume = {10},
number = {1},
pages = {3--24},
DOI = {10.2478/s11533-011-0112-1},
author = {Schweitzer, Marc Alexander}
}
@Article { Schweitzer:2012,
title = {Generalizations of the Finite Element Method},
journal = {Central European Journal of Mathematics},
year = {2012},
volume = {10},
pages = {3-24},
publisher = {SP Versita},
DOI = {10.2478/s11533-011-0112-1},
author = {Schweitzer, Marc Alexander}
}
@Article { Schweitzer:2011,
title = {Multilevel Particle--Partition of Unity Method},
journal = {Numer. Math.},
year = {2011},
volume = {118},
pages = {307--328},
author = {M.\verb=~=A.\verb=~=Schweitzer}
}
@Article { MR2800711,
title = {Multilevel particle-partition of unity method},
journal = {Numer. Math.},
year = {2011},
volume = {118},
number = {2},
pages = {307--328},
DOI = {10.1007/s00211-010-0346-z},
author = {Schweitzer, Marc Alexander}
}
@Article { Schweitzer:2011*1,
title = {Stable Enrichment and Local Preconditioning in the Particle--Partition of Unity Method},
journal = {Numer. Math.},
year = {2011},
volume = {118},
number = {1},
pages = {137--170},
author = {M.\verb=~=A.\verb=~=Schweitzer}
}
@Article { MR2793905,
title = {Stable enrichment and local preconditioning in the particle-partition of unity method},
journal = {Numer. Math.},
year = {2011},
volume = {118},
number = {1},
pages = {137--170},
DOI = {10.1007/s00211-010-0323-6},
author = {Schweitzer, Marc Alexander}
}
@Incollection { MR3220498,
title = {Treatment of general domains in two space dimensions in a partition of unity method},
year = {2011},
volume = {79},
pages = {27--49},
publisher = {Springer, Heidelberg},
series = {Lect. Notes Comput. Sci. Eng.},
booktitle = {Meshfree methods for partial differential equations V},
DOI = {10.1007/978-3-642-16229-9_2},
author = {Schweitzer, Marc Alexander and Randrianarivony, Maharavo}
}
@Article { MR2643198,
title = {Numerical simulation of bubble and droplet deformation by a level set approach with surface tension in three dimensions},
journal = {Internat. J. Numer. Methods Fluids},
year = {2010},
volume = {62},
number = {9},
pages = {963--993},
DOI = {10.1002/fld.2051},
author = {Croce, Roberto and Griebel, Michael and Schweitzer, Marc Alexander}
}
@Article { Schweitzer:2009,
title = {An Adaptive hp-Version of the Multilevel Particle--Partition of Unity Method},
journal = {Comput. Methods Appl. Mech. Engrg.},
year = {2009},
volume = {198},
pages = {1260--1272},
DOI = {10.1016/j.cma.2008.01.009},
author = {M.A.\verb=~=Schweitzer}
}
@Article { MR2500246,
title = {An adaptive hp-version of the multilevel particle-partition of unity method},
journal = {Comput. Methods Appl. Mech. Engrg.},
year = {2009},
volume = {198},
number = {13-14},
pages = {1260--1272},
DOI = {10.1016/j.cma.2008.01.009},
author = {Schweitzer, Marc Alexander}
}
@Article { Schweitzer:2009*1,
title = {An Algebraic Treatment of Essential Boundary Conditions in the Particle--Partition of Unity Method},
journal = {SIAM Journal on Scientific Computing},
year = {2009},
volume = {31},
number = {2},
pages = {1581-1602},
DOI = {10.1137/080716499},
author = {M.\verb=~=A.\verb=~=Schweitzer}
}
@Article { Croce.Griebel.Schweitzer:2009,
title = {Numerical Simulation of Bubble and Droplet-Deformation by a Level Set Approach with Surface Tension in Three Dimensions},
journal = {International Journal for Numerical Methods in Fluids},
year = {2009},
volume = {62},
number = {9},
pages = {963--993},
note = {Also available as SFB 611 Preprint no 431},
DOI = {10.1002/fld.2051},
author = {R.\verb=~=Croce and M.\verb=~=Griebel and M.\verb=~=A.\verb=~=Schweitzer}
}
@Article { MR2486844,
title = {An algebraic treatment of essential boundary conditions in the particle-partition of unity method},
journal = {SIAM J. Sci. Comput.},
year = {2008},
volume = {31},
number = {2},
pages = {1581--1602},
DOI = {10.1137/080716499},
author = {Schweitzer, Marc Alexander}
}
@Inproceedings { Griebel.Metsch.Schweitzer:2008,
title = {Coarse Grid Classification: AMG on Parallel Computers},
year = {2008},
volume = {39},
pages = {299--306},
note = {Also available as SFB 611 preprint No. 368, Universit{\~A}¤t Bonn, 2008},
editor = {Gernot M{\~A}¼nster and Dietrich Wolf and Manfred Kremer},
series = {NIC Series},
booktitle = {NIC Symposium 2008},
author = {Griebel, Michael and Metsch, Bram and Alexander Schweitzer, Marc}
}
@Phdthesis { Schweitzer:2008_2,
title = {Meshfree and Generalized Finite Element Methods},
year = {2008},
school = {Institute for Numerical Simulation, University of Bonn},
type = {Habilitation},
author = {A. Schweitzer, M.}
}
@Article { Griebel.Scherer.Schweitzer:2007,
title = {Robust Norm Equivalencies for Diffusion Problems},
journal = {Math. Comput.},
year = {2007},
volume = {76},
pages = {1141--1161},
author = {M.\verb=~=Griebel and K.\verb=~=Scherer and M.\verb=~=A.\verb=~=Schweitzer}
}
@Article { MR2299769,
title = {Robust norm equivalencies for diffusion problems},
journal = {Math. Comp.},
year = {2007},
volume = {76},
number = {259},
pages = {1141--1161},
DOI = {10.1090/S0025-5718-07-01973-4},
author = {Griebel, Michael and Scherer, Karl and Schweitzer, Alexander}
}
@Incollection { MR2293516,
title = {A particle-partition of unity method. VII. Adaptivity},
year = {2007},
volume = {57},
pages = {121--147},
publisher = {Springer, Berlin},
series = {Lect. Notes Comput. Sci. Eng.},
booktitle = {Meshfree methods for partial differential equations III},
DOI = {10.1007/978-3-540-46222-4_8},
author = {Griebel, Michael and Schweitzer, Marc Alexander}
}
@Article { Griebel.Metsch.Oeltz.ea:2006,
title = {Coarse Grid Classification: A Parallel Coarsening Scheme For Algebraic Multigrid Methods},
journal = {Numerical Linear Algebra with Applications},
year = {2006},
volume = {13},
number = {2--3},
pages = {193--214},
abstract = {In this paper we present a new approach to theparallelization of algebraic multigrid (AMG), i.e., to theparallel coarse grid selection in AMG. Our approach doesnot involve any special treatment of processor subdomainboundaries and hence avoids a number of drawbacks of otherAMG parallelization techniques. The an appropriate (local)coarse grid on each processor from all admissible gridssuch that the composed coarse grid forms a suitable coarsegrid for the whole domain, i.e. there is no need for anyboundary treatment. To this end, we first constructmultiple equivalent coarse grids on each processorsubdomain. In a second step we then select exactly one gridper processor by a graph clustering technique. The resultsof our numerical experiments clearly indicate that thisapproach results in coarse grids of high quality which arevery close to those obtained with sequential AMG.Furthermore, the operator and grid complexities of ourparallel AMG are mostly smaller than those obtained byother parallel AMG methods, whereas the scale-up behaviorof the proposed algorithm is similar to that of otherparallel AMG techniques. However a significant improvementwith respect to the speed-up performance is achieved.},
note = {Also available as SFB 611 preprint No. 225, Universit{\~A}¤t Bonn, 2005},
author = {M.\verb=~=Griebel and B\verb=~=.Metsch and D\verb=~=.Oeltz and M.\verb=~=A.\verb=~=Schweitzer}
}
@Article { MR2209433,
title = {Coarse grid classification: a parallel coarsening scheme for algebraic multigrid methods},
journal = {Numer. Linear Algebra Appl.},
year = {2006},
volume = {13},
number = {2-3},
pages = {193--214},
DOI = {10.1002/nla.482},
author = {Griebel, Michael and Metsch, Bram and Oeltz, Daniel and Schweitzer, Marc Alexander}
}
@Incollection { MR2278264,
title = {A particle-partition of unity method. VI. A p-robust multilevel solver},
year = {2005},
volume = {43},
pages = {71--92},
publisher = {Springer, Berlin},
series = {Lect. Notes Comput. Sci. Eng.},
booktitle = {Meshfree methods for partial differential equations II},
DOI = {10.1007/3-540-27099-X_5},
author = {Griebel, Michael and Oswald, Peter and Schweitzer, Marc Alexander}
}
@Incollection { MR2178272,
title = {Efficient implementation and parallelization of meshfree and particle methods---the parallel multilevel partition of unity method},
year = {2005},
pages = {195--262},
publisher = {Springer, Berlin},
series = {Universitext},
booktitle = {Frontiers of numerical analysis},
DOI = {10.1007/3-540-28884-8_4},
author = {Schweitzer, Marc Alexander}
}
@Article { Clarenz.Griebel.Rumpf.ea:2004,
title = {Feature Sensitive Multiscale Editing on Surfaces},
journal = {The Visual Computer},
year = {2004},
number = {20},
pages = {329--343},
note = {Also as Preprint No.\verb=~=89, SFB 611, Universit{\~A}¤t Bonn, Germany},
author = {Clarenz, U. and Griebel, M. and Rumpf, M. and Schweitzer, M. and Telea, A.}
}
@Article { Griebel.Oeltz.Schweitzer:2003,
title = {An Algebraic Multigrid Method for Linear Elasticity},
journal = {SIAM Journal on Scientific Computing},
year = {2003},
volume = {25},
number = {2},
pages = {385--407},
author = {M.\verb=~=Griebel and D.\verb=~=Oeltz and M.\verb=~=A.\verb=~=Schweitzer}
}
@Article { MR2058066,
title = {An algebraic multigrid method for linear elasticity},
journal = {SIAM J. Sci. Comput.},
year = {2003},
volume = {25},
number = {2},
pages = {385--407},
DOI = {10.1137/S1064827502407810},
author = {Griebel, Michael and Oeltz, Daniel and Schweitzer, Marc Alexander}
}
@Book { MR1988237,
title = {A parallel multilevel partition of unity method for elliptic partial differential equations},
year = {2003},
volume = {29},
pages = {iv+194},
publisher = {Springer-Verlag, Berlin},
series = {Lecture Notes in Computational Science and Engineering},
ISBN = {3-540-00351-7},
DOI = {10.1007/978-3-642-59325-3},
author = {Schweitzer, Marc Alexander}
}
@Book { Schweitzer:2003*4,
title = {A Parallel Multilevel Partition of Unity Method for Elliptic Partial Differential Equations},
year = {2003},
volume = {29},
publisher = {Springer},
series = {Lecture Notes in Computational Science and Engineering},
author = {M.\verb=~=A.\verb=~=Schweitzer}
}
@Incollection { MR2003437,
title = {A particle-partition of unity method. IV. Parallelization},
year = {2003},
volume = {26},
pages = {161--192},
publisher = {Springer, Berlin},
series = {Lect. Notes Comput. Sci. Eng.},
booktitle = {Meshfree methods for partial differential equations (Bonn, 2001)},
DOI = {10.1007/978-3-642-56103-0_12},
author = {Griebel, Michael and Schweitzer, Marc Alexander}
}
@Incollection { MR2008354,
title = {A particle-partition of unity method. V. Boundary conditions},
year = {2003},
pages = {519--542},
publisher = {Springer, Berlin},
booktitle = {Geometric analysis and nonlinear partial differential equations},
author = {Griebel, M. and Schweitzer, M. A.}
}
@Article { Griebel.Schweitzer:2002*2,
title = {A Particle-Partition of Unity Method---Part II: Efficient Cover Construction and Reliable Integration},
journal = {SIAM Journal on Scientific Computing},
year = {2002},
volume = {23},
number = {5},
pages = {1655--1682},
abstract = {In this paper we present a meshfree discretizationtechnique based only on a set of irregularly spaced pointsand the partition of unity approach. We focus on the coverconstruction and its interplay with the integration problemarising in a Galerkin discretization. We present ahierarchical cover construction algorithm and a reliabledecomposition quadrature scheme. Here, we decompose theintegration domains into disjoint cells on which we employlocal sparse grid quadrature rules to improve computationalefficiency. The use of these two schemes already reducesthe operation count for the assembly of the stiffnessmatrix significantly. Now, the overall computational costsare dominated by the number of the integration cells. Wepresent a regularized version of the hierarchical coverconstruction algorithm which reduces the number ofintegration cells even further and subsequently improvesthe computational efficiency. In fact, the computationalcosts during the integration of the nonzeros of thestiffness matrix are comparable to that of a finite elementmethod, yet the presented method is completely independentof a mesh. Moreover, our method is applicable to generaldomains and allows for the construction of approximationsof any order and regularity.},
author = {M.\verb=~=Griebel and Schweitzer, M.\verb=~=A.}
}
@Article { Griebel.Schweitzer:2002*4,
title = {A Particle-Partition of Unity Method---Part III: A Multilevel Solver},
journal = {SIAM Journal on Scientific Computing},
year = {2002},
volume = {24},
number = {2},
pages = {377--409},
abstract = {In this paper we focus on the efficient solution of thelinear block-systems arising from a Galerkin discretizationof an elliptic partial differential equation of secondorder with the partition of unity method (PUM). We presenta cheap multilevel solver for partition of unitydiscretizations of any order. The shape functions of a PUMare products of piecewise rational partition of unity (PU)functions and higher order local approximation functions(usually a local polynomial). Furthermore, they arenon-interpolatory. In a multilevel approach we not onlyhave to cope with non-interpolatory basis functions butalso with a sequence of nonnested spaces due to themeshfree construction. Hence, injection or interpolatoryinterlevel transfer operators are not available for ourmultilevel PUM. Therefore, the remaining natural choice forthe prolongation operators are L2-projections. Here, weexploit the partition of unity construction of the functionspaces and a hierarchical construction of the PU itself tolocalize the corresponding projection problem. Thissignificantly reduces the computational costs associatedwith the setup and the application of the interleveltransfer operators. The second main ingredient for ourmultilevel solver is the use of a block-smoother to treatthe local approximation functions simultaneously. Theresults of our numerical experiments in two and threedimensions show that the convergence rate of the proposedmultilevel solver is independent of the number of patches.The convergence rate is slightly dependent on the localapproximation orders.},
institution = {Sonderforschungsbereich 256, Institut f{\~A}¼r Angewandte Mathematik, Universit{\~A}¤t Bonn},
author = {M.\verb=~=Griebel and Schweitzer, M.\verb=~=A.}
}
@Article { MR1885078,
title = {A particle-partition of unity method. II. Efficient cover construction and reliable integration},
journal = {SIAM J. Sci. Comput.},
year = {2002},
volume = {23},
number = {5},
pages = {1655--1682},
DOI = {10.1137/S1064827501391588},
author = {Griebel, Michael and Schweitzer, Marc Alexander}
}
@Article { MR1951047,
title = {A particle-partition of unity method. III. A multilevel solver},
journal = {SIAM J. Sci. Comput.},
year = {2002},
volume = {24},
number = {2},
pages = {377--409},
DOI = {10.1137/S1064827501395252},
author = {Griebel, Michael and Schweitzer, Marc Alexander}
}
@Article { Griebel.Schweitzer:2000,
title = {A Particle-Partition of Unity Method for the Solution of Elliptic, Parabolic and Hyperbolic PDE},
journal = {SIAM Journal on Scientific Computing},
year = {2000},
volume = {22},
number = {3},
pages = {853--890},
abstract = {In this paper, we present a meshless discretizationtechnique for instationary convec\textbackslash-tion-diffusion problems.It is based on operator splitting, the method ofcharacteristics and a generalized partition of unitymethod. We focus on the discretization process and itsquality. The method may be used as an h- or p-version. Evenfor general particle distributions, the convergencebehavior of the different versions corresponds to that ofthe respective version of the finite element method on auniform grid. We discuss the implementational aspects ofthe proposed method. Furthermore, we present the results ofnumerical examples, where we considered instationaryconvection-diffusion, instationary diffusion, linearadvection and elliptic problems.},
author = {M.\verb=~=Griebel and M.\verb=~=A.\verb=~=Schweitzer}
}
@Article { MR1785338,
title = {A particle-partition of unity method for the solution of elliptic, parabolic, and hyperbolic PDEs},
journal = {SIAM J. Sci. Comput.},
year = {2000},
volume = {22},
number = {3},
pages = {853--890},
DOI = {10.1137/S1064827599355840},
author = {Griebel, Michael and Schweitzer, Marc Alexander}
}
@Inproceedings { Caglar.Griebel.Schweitzer.ea:1999,
title = {Dynamic Load-Balancing of Hierarchical Tree Algorithms on a Cluster of Multiprocessor PCs and on the Cray T3E},
year = {1999},
abstract = {The solution of many problems in science and engineering is based on computational kernels for the numerical treatment of partial differential equations (PDEs) or N-body problems. Traditional solution methods however reduce these to linear algebra or brute force algorithms on structured data sets. Larger and larger simulations require smarter algorithms to be tractable. Hierarchical tree algorithms represent such a class, both for PDEs and for N-body problems. However, their efficient parallelization is not straightforward. Some difficulties can be removed, if one can provide a fast dynamic load-balancing scheme to cope with the tree variations of the unstructured data sets. In this paper we propose a very cheap yet very efficient load-balancing scheme for tree algorithms based on space-filling curves. Furthermore we present the Parnass2 cluster, on which such parallel codes perform extremely well. The cluster consists of SMP PCs and a Myrinet network at Gigabit/s speed configured with full bisection bandwidth. It turns out that it does not only has the obvious price/performance advantage, but also an absolute performance, which is comparable to the latest commercial Cray T3E.},
note = {SuParCup '99 Award Winning Paper, also as SFB 256 report 27},
editor = {H. W. Meuer},
publisher = {Mateo},
address = {Mannheim, Germany},
series = {ISBN 3-932178-08-4},
booktitle = {Proceedings 14th Supercomputer Conference, Mannheim},
author = {Caglar, A. and Griebel, M. and A. Schweitzer, M. and Zumbusch, G.}
}
@Inproceedings { Schweitzer.Zumbusch.Griebel:1999,
title = {Parnass2: A Cluster of Dual-Processor PCs},
year = {1999},
number = {CSR-99-02},
pages = {45--54},
abstract = {We report on a cluster with 96 CPUs (48 Dual-Processor PCs running Linux 2.1.127) at our department, called Parnass2. The computing nodes are connected by a Myrinet, a Gigabit per second switched LAN, and additionally by a Fast-Ethernet. A comparison of different message passing (MPI) libraries for the Myrinet is given. Furthermore, we present the performance results of Parnass2 for some of the parallel codes developed at our department, namely a sparse grid code for PDEs, a particle code for molecular dynamics, a finite difference code for the Navier-Stokes equation, and a parallel adaptive finite element / finite difference multi-grid code. We compare these results with the performance of these codes running on a Cray T3E-1200, a Cray T3E-600, a SGI Origin 200/2000 and Parnass \textbackslashciteMGriebel:GWZumbusch:1997a, a cluster of SGI O2 workstations.},
editor = {W. Rehm and T. Ungerer},
address = {Chemnitz, Germany},
series = {Chemnitzer Informatik Berichte},
booktitle = {Proceedings of the 2nd Workshop Cluster-Computing, Karlsruhe},
organization = {TU Chemnitz},
author = {A. Schweitzer, M. and W. Zumbusch, G. and Griebel, M.}
}
@Mastersthesis { Schweitzer:1997,
title = {Ein Partikel-Galerkin-Verfahren mit Ansatzfunktionen der Partition of Unity Method},
year = {1997},
school = {Institut f{\~A}¼r Angewandte Mathematik, Universit{\~A}¤t Bonn},
type = {Diplomarbeit},
author = {Schweitzer, M. A.}
}