The numerical treatment of partial differential equations with meshfree discretization techniques has been a very active research area in recent years. Up to now, however, meshfree methods have been in an early experimental stage and were not competitive due to the lack of efficient iterative solvers and numerical quadrature. This volume now presents an efficient parallel implementation of a meshfree method, namely the partition of unity method (PUM). A general numerical integration scheme is presented for the efficient assembly of the stiffness matrix as well as an optimal multilevel solver for the arising linear system. Furthermore, detailed information on the parallel implementation of the method on distributed memory computers is provided and numerical results are presented in two and three space dimensions with linear, higher order and augmented approximation spaces with up to 42 million degrees of freedom
CONTENT
1 Introduction -- 2 Partition of Unity Method -- 2.1 Construction of a Partition of Unity Space -- 2.2 Properties -- 2.3 Basic Convergence Theory -- 3 Treatment of Elliptic Equations -- 3.1 Galerkin Discretization -- 3.2 Boundary Conditions -- 3.3 Numerical Results -- 4 Multilevel Solution of the Resulting Linear System -- 4.1 Multilevel Iterative Solvers -- 4.2 Multilevel Partition of Unity Method -- 4.3 Numerical Results -- 5 Tree Partition of Unity Method -- 5.1 Single Level Cover Construction -- 5.2 Construction of a Sequence of PUM Spaces -- 5.3 Numerical Results -- 6 Parallelization and Implementational Details -- 6.1 Parallel Data Structures -- 6.2 Parallel Tree Partition of Unity Method -- 6.3 Numerical Results -- 7 Concluding Remarks -- Treatment of other Types of Equations -- A.1 Parabolic Equations -- A.2 Hyperbolic Equations -- Transformation of Keys -- Color Plates -- References
Mathematics
Mathematical analysis
Analysis (Mathematics)
Partial differential equations
Computer mathematics
Physics
Applied mathematics
Engineering mathematics
Mathematics
Analysis
Computational Mathematics and Numerical Analysis
Numerical and Computational Physics
Partial Differential Equations
Appl.Mathematics/Computational Methods of Engineering
