By Marc Alexander Schweitzer
The numerical remedy of partial differential equations with meshfree discretization ideas has been a really lively study zone lately. during the past, although, meshfree equipment 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 offers an effective parallel implementation of a meshfree approach, particularly the partition of cohesion procedure (PUM). A common 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 method. in addition, exact details at the parallel implementation of the tactic on allotted reminiscence desktops is supplied and numerical effects are provided in and 3 area dimensions with linear, larger order and augmented approximation areas with as much as forty two million levels of freedom.
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Extra info for A Parallel Multilevel Partition of Unity Method for Elliptic Partial Differential Equations
Further experiments with our PUM with quadratic and cubic splines as weight functions in the Shepard construction gave very much the same approximation results. Hence, the presented optimal convergence rates we obtain with our PUM do not only hold for CO-discretizations but also for smoother approximations in Ct with t 2: o. Chapter 4 Multilevel Solution of the Resulting Linear System In the following we focus on the solution of the large sparse linear blocksystem Au = j where u denotes a coefficient vector and j denotes a moment vector.
3. Quadrature nodes of two sparse grid Gauss-Patterson rules, level 1 = 6 with 769 nodes in two dimension (left) and level 1 = 5 with 1023 nodes in three dimension (right) . , lEN and k E Nd . Due to the restriction in the summation, the number of quadrature points sparse grid quadrature rule Q1 is n1 L:t=l ki x1 ::; l + d - 1 of the resulting only. 3) than for a full tensor product rule where the integrand has to be evaluated at O(21d) quadrature points. This reduction of the computational costs, however, does not compromise the approximation quality significantly for smooth functions.
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. 3. 1. 1. 500 by increasing the refinement level l of the underlying uniform grid.