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 timedependent settings.
The particlepartition 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 socalled 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 wellestablished 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.


[ 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 : 93110 Publisher: Springer 2013[ 4] M.~A.~Schweitzer , S.~Wu
Numerical Integration of ontheflycomputed 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 ParticlePartition of Unity Method for the Solution of Elliptic, Parabolic and Hyperbolic PDE SIAM Journal on Scientific Computing , 22: (3): 853890 2000[ 6] M.~Griebel , M.~A. Schweitzer
A ParticlePartition of Unity MethodPart II: Efficient Cover Construction and Reliable Integration SIAM Journal on Scientific Computing , 23: (5): 16551682 2002[ 7] M.~Griebel , M.~A. Schweitzer
A ParticlePartition of Unity MethodPart III: A Multilevel Solver SIAM Journal on Scientific Computing , 24: (2): 377409 2002[ 8] M.A.~Schweitzer
An Adaptive hpVersion of the Multilevel ParticlePartition of Unity Method Comput. Methods Appl. Mech. Engrg. , 198: : 12601272 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): 385407 2003[ 10] M.~A.~Schweitzer
An Algebraic Treatment of Essential Boundary Conditions in the ParticlePartition of Unity Method SIAM Journal on Scientific Computing , 31: (2): 15811602 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, UniversitÃ¤t Bonn, 2005 Numerical Linear Algebra with Applications , 13: (23): 193214 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, UniversitÃ¤t Bonn, Germany The Visual Computer (20): 329343 2004[13] Marc Alexander Schweitzer
Generalizations of the Finite Element Method Central European Journal of Mathematics , 10: : 324 Publisher: SP Versita 2012 DOI: 10.2478/s1153301101121 [ 14] M.~A.~Schweitzer
Multilevel ParticlePartition of Unity Method Numer. Math. , 118: : 307328 2011[ 15] R.~Croce , M.~Griebel , M.~A.~Schweitzer
Numerical Simulation of Bubble and DropletDeformation 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): 963993 2009 DOI: 10.1002/fld.2051[ 17] M.~A.~Schweitzer
Stable Enrichment and Local Preconditioning in the ParticlePartition of Unity Method Numer. Math. , 118: (1): 137170 2011[ 18] M. Schweitzer
Variational Mass Lumping in the Partition of Unity Method SIAM Journal on Scientific Computing , 35: (2): A1073A1097 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 2003

