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

By Marc Alexander Schweitzer

The numerical therapy of partial differential equations with meshfree discretization innovations has been a truly energetic study sector in recent times. prior to now, besides the fact that, meshfree tools were in an early experimental degree and weren't aggressive a result of loss of effective iterative solvers and numerical quadrature. This quantity now offers an effective parallel implementation of a meshfree process, specifically the partition of team spirit procedure (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 coming up linear procedure. moreover, distinctive details at the parallel implementation of the strategy on dispensed reminiscence desktops is equipped and numerical effects are provided in and 3 house dimensions with linear, greater order and augmented approximation areas with as much as forty two million levels of freedom.

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2. ij of the mapping T~ij : [-l,l]d --+ D~ij from the reference integration domain [-1, l]d onto the integration cell D~ij in the configuration space. 1 3. 3) on the integration cells D~ ... That is, we compute them on the reference integration domain " [-1, l]d using the transformations T~ij and the respective Jacobians JT.! 1 " Just like in the FEM or GFEM, these mappings can for instance be constructed by the blending function method [136]. To this end, it may be necessary to refine the parametric integration cells D~ij depending on their geometric complexity; see [134] for a detailed discussion of this subject in the GFEM context.

Nonzeros(IZ_1) k-1' d C(I k-1) '= nonzeros(I;-l) k' dofk dofk ' • These averages C(A k ), C(IL 1) and C(I;-1) are also the relevant measures for the storage requirement of the method. With these estimates for the matrices we can estimate the operation counts per unknown coefficient on level k. 1. Multilevel Iterative Solvers CD,k = 2C(I;-1) + 2C(A k ) + 1 and 55 CO,k = 1 + 2C(ILl) where 2C(I;-1), 2C(ILl) and 2C(A k ) denote the respective costs of the matrix-vector-product (one multiplication and summation per coefficient).

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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