A Parallel Multilevel Partition of Unity Method for Elliptic by Marc Alexander Schweitzer

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By Marc Alexander Schweitzer

The numerical therapy of partial differential equations with meshfree discretization options has been a really lively study zone lately. in past times, despite the fact that, meshfree tools were in an early experimental level and weren't aggressive end result of the loss of effective iterative solvers and numerical quadrature. This quantity now provides a good parallel implementation of a meshfree technique, specifically the partition of harmony strategy (PUM). A basic numerical integration scheme is gifted for the effective meeting of the stiffness matrix in addition to an optimum multilevel solver for the bobbing up linear procedure. in addition, precise details at the parallel implementation of the tactic on disbursed reminiscence desktops is equipped and numerical effects are awarded in and 3 house dimensions with linear, larger order and augmented approximation areas with as much as forty two million levels of freedom.

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For our PVM space we have dof ~ Npd where N = card(Cn) denotes the number of patches Wi and p = maxi Pi. Hence, the number of operations necessary to solve the stiffness matrix with a classical direct solver is of the order O((Np d)3). Since the stiffness matrix A is a sparse blockmatrix with dense blocks its storage demand is ofthe order O(Np2d), yet the storage requirement of the method would increase to O((Np d)2) if we apply a direct solver. The use of a more advanced direct solver for sparse matrices can cure this dramatic increase in compute time and storage requirements to some extent only.

The eigenvalue Amax can be computed very efficiently by a simultaneous Rayleigh-quotient minimization method [17, 46, 96] due to the similar structure of the matrices A and B. On average we need about five to ten conjugate gradient iterations to compute Amax with five digits accuracy. Here, the minimization of ~; ~~ involves only matrix-vector-products. We do not need to solve a linear system. 10) involves only boundary degrees of freedom and is therefore of smaller dimension. Hence, the computational costs associated with the assembly of a Dirichlet problem are comparable to the costs associated with the respective Neumann problem.

I..... . "....... , .. .... .. . ........ ................... . • I • •• J ..... • I • I • .. o.. , ••••• 'e ' •• . • . • .................. ". · · -,--,, -, - -,- ' ""' . - --.. e . • . • .. 1 . . . I.. •. :. :.. :. : • ....... ~ •• \, • • • ~ •• \,. • • •• . ~~ .. ~ ~~ .. ~ ~. ~.. .. - - - -... --. : :- :: . 5. 5 (right) . The support of a single shape function is indicated by the gray shaded area. 6. 1. See Color Plate 1 on page 173. 1. 6. , we use Pi = P = 1. 3. The number of degrees of freedom dof is given by 3N where N = card(P) = card(Cn) denotes the number of points (or cover patches) in this two-dimensional example.

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