Download PDF by Marc Alexander Schweitzer: A Parallel Multilevel Partition of Unity Method for Elliptic
By Marc Alexander Schweitzer
The numerical therapy of partial differential equations with meshfree discretization recommendations has been a really energetic learn quarter lately. in the past, despite the fact that, meshfree equipment were in an early experimental level and weren't aggressive a result of loss of effective iterative solvers and numerical quadrature. This quantity now offers a good parallel implementation of a meshfree strategy, particularly the partition of cohesion approach (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 method. additionally, distinctive details at the parallel implementation of the tactic on disbursed reminiscence desktops is supplied and numerical effects are offered in and 3 area dimensions with linear, greater order and augmented approximation areas with as much as forty two million levels of freedom.
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Additional resources for A Parallel Multilevel Partition of Unity Method for Elliptic Partial Differential Equations
This loss of generality and freedom can hardly be justified. Furthermore, the coupling itself must be consistent with the local approximation orders Pi of the local spaces or we also experience an adverse effect on the approximation quality of the overall method only due to the poor implementation of Dirichlet boundary conditions. The penalty or perturbation approaches are very general concepts for the implementation of constraints in a variational problem. 2. Boundary Conditions 33 we would introduce an additional surface term in the variational formulation to enforce the boundary conditions.
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. 3. 1. 1. 500 by increasing the refinement level l of the underlying uniform grid.
6. , we may include singular functions in a local approximation space without the need to pay any attention to neighboring (overlapping) local spaces. Hence, the introduction of a singular function into the local approximation spaces V,Pi in the vicinity of the re-entrant corner can be realized very easily within the PUM. 3. 5. 5. 14. 5 (left) for an L-shaped domain D. A partition of D into the sub-domains Dp (white) where we use the standard local i and Da (gray shaded) where augmented local approxiapproximation spaces mation spaces 1f;pf are used (right).
A Parallel Multilevel Partition of Unity Method for Elliptic Partial Differential Equations by Marc Alexander Schweitzer