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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.

Differential equations, Elliptic --- Partition of unity method. --- Numerical solutions --- Data processing. --- -681.3*G18 --- 519.63 --- Partition of unity method --- PUM (Numerical analysis) --- Meshfree methods (Numerical analysis) --- 519.63 Numerical methods for solution of partial differential equations --- Numerical methods for solution of partial differential equations --- 681.3 *G18 Partial differential equations: difference methods; elliptic equations; finite element methods; hyperbolic equations; method of lines; parabolic equations (Numerical analysis) --- Partial differential equations: difference methods; elliptic equations; finite element methods; hyperbolic equations; method of lines; parabolic equations (Numerical analysis) --- Elliptic differential equations --- Elliptic partial differential equations --- Linear elliptic differential equations --- Differential equations, Linear --- Differential equations, Partial --- -Data processing --- 681.3*G18 --- Numerical solutions&delete& --- Data processing --- 681.3 *G18 --- Mathematical analysis. --- Analysis (Mathematics). --- Computer mathematics. --- Physics. --- Partial differential equations. --- Applied mathematics. --- Engineering mathematics. --- Analysis. --- Computational Mathematics and Numerical Analysis. --- Numerical and Computational Physics, Simulation. --- Partial Differential Equations. --- Mathematical and Computational Engineering. --- Engineering --- Engineering analysis --- Mathematical analysis --- Partial differential equations --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Computer mathematics --- Electronic data processing --- Mathematics --- 517.1 Mathematical analysis