The development of efficient meshfree methods for partial differential equations is at the center of my research work. Since these modern methods do not require the availability of an appropriate computational mesh they alleviate the treatment of applications in complex domains, especially in time-dependent settings.
The particle-partition of unity method, a meshfree generalization of the finite element method I have developed together with Prof. Dr. M. Griebel, furthermore allows for the easy incorporation of a priori information about special local behavior of the solution (e.g. discontinuities and singularities) by so-called enrichment. If such information is available (analytically or numerically) this enrichment technique reduces the computational complexity substantially since it eliminates the need for classical adaptive refinement. This enrichment technique can also be interpreted as a multiscale coupling.
In fracture mechanics for instance good enrichment information is often available from asymptotic expansions of the solution. Hence, the use of enriched approximation techniques is becoming well-established in this field. In general however the numerical construction of appropriate enrichment information is necessary. Here, the use of adaptive refinement or microscale simulations must be employed. The automatic computation of fine scale enrichment information and its incorporation in a coarse scale simulation is a very important research topic with wide application and thus shall be further explored also in the future.
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[ 1] M.~A.~Schweitzer , A.~Ziegenhagel
Dispersion Properties of the Partition of Unity Method & Explicit Dynamics
Meshfree Methods for Partial Differential Equations VII
Also available as INS Preprint No. 1407
of Lecture Notes in Computational Science and Engineering
Publisher: Springer
2015[ 2] P.~Diehl , M.~A.~Schweitzer
Efficient Neighbor Search for Particle Methods on GPUs
Meshfree Methods for Partial Differential Equations VII
Also available as INS Preprint No. 1405
of Lecture Notes in Computational Science and Engineering
Publisher: Springer
2015[ 3] M.~A.~Schweitzer
Multilevel Partition of Unity Method for Elliptic Problems with Strongly Discontinuous Coefficients
Meshfree Methods for Partial Differential Equations VI
of Lecture Notes in Computational Science and Engineering : 93--110
Publisher: Springer
2013[ 4] M.~A.~Schweitzer , S.~Wu
Numerical Integration of on-the-fly-computed Enrichment Functions in the PUM
Meshfree Methods for Partial Differential Equations VII
Also available as INS Preprint No. 1406
of Lecture Notes in Computational Science and Engineering
Publisher: Springer
2015[ 5] M.~Griebel , M.~A.~Schweitzer
A Particle-Partition of Unity Method for the Solution of Elliptic, Parabolic and Hyperbolic PDE
SIAM Journal on Scientific Computing , 22: (3): 853--890
2000[ 6] M.~Griebel , M.~A. Schweitzer
A Particle-Partition of Unity Method---Part II: Efficient Cover Construction and Reliable Integration
SIAM Journal on Scientific Computing , 23: (5): 1655--1682
2002[ 7] M.~Griebel , M.~A. Schweitzer
A Particle-Partition of Unity Method---Part III: A Multilevel Solver
SIAM Journal on Scientific Computing , 24: (2): 377--409
2002[ 8] M.A.~Schweitzer
An Adaptive hp-Version of the Multilevel Particle--Partition of Unity Method
Comput. Methods Appl. Mech. Engrg. , 198: : 1260--1272
2009
DOI: 10.1016/j.cma.2008.01.009[ 9] M.~Griebel , D.~Oeltz , M.~A.~Schweitzer
An Algebraic Multigrid Method for Linear Elasticity
SIAM Journal on Scientific Computing , 25: (2): 385--407
2003[ 10] M.~A.~Schweitzer
An Algebraic Treatment of Essential Boundary Conditions in the Particle--Partition of Unity Method
SIAM Journal on Scientific Computing , 31: (2): 1581-1602
2009
DOI: 10.1137/080716499[ 11] M.~Griebel , B~.Metsch , D~.Oeltz , M.~A.~Schweitzer
Coarse Grid Classification: A Parallel Coarsening Scheme For Algebraic Multigrid Methods
Also available as SFB 611 preprint No. 225, Universität Bonn, 2005
Numerical Linear Algebra with Applications , 13: (2--3): 193--214
2006[ 12] U. Clarenz, M. Griebel, M. Rumpf, M. Schweitzer, A. Telea
Feature Sensitive Multiscale Editing on Surfaces
Also as Preprint No.~89, SFB 611, Universität Bonn, Germany
The Visual Computer (20): 329--343
2004[ 13] Marc Alexander Schweitzer
Generalizations of the Finite Element Method
Central European Journal of Mathematics , 10: : 3-24
Publisher: SP Versita
2012
DOI: 10.2478/s11533-011-0112-1[ 14] M.~A.~Schweitzer
Multilevel Particle--Partition of Unity Method
Numer. Math. , 118: : 307--328
2011[ 15] R.~Croce , M.~Griebel , M.~A.~Schweitzer
Numerical Simulation of Bubble and Droplet-Deformation by a Level Set Approach with Surface Tension in Three Dimensions
Also available as SFB 611 Preprint no 431
International Journal for Numerical Methods in Fluids , 62: (9): 963--993
2009
DOI: 10.1002/fld.2051[ 17] M.~A.~Schweitzer
Stable Enrichment and Local Preconditioning in the Particle--Partition of Unity Method
Numer. Math. , 118: (1): 137--170
2011[ 18] M. Schweitzer
Variational Mass Lumping in the Partition of Unity Method
SIAM Journal on Scientific Computing , 35: (2): A1073-A1097
2013
DOI: 10.1137/120895561[ 19] M.~A.~Schweitzer
A Parallel Multilevel Partition of Unity Method for Elliptic Partial Differential Equations
of Lecture Notes in Computational Science and Engineering
Publisher: Springer
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