In this sequel to [12, 13, 14, 15] we focus on the implementation of Dirichlet boundary conditions in our partition of unity method. The treatment of essential boundary conditions with meshfree Galerkin methods is not an easy task due to the non-interpolatory character of the shape functions. Here, the use of an almost forgotten method due to Nitsche from the 1970's allows us to overcome these problems at virtually no extra computational costs. The method is applicable to general point distributions and leads to positive definite linear systems. The results of our numerical experiments, where we consider discretizations with several million degrees of freedom in two and three dimensions, clearly show that we achieve the optimal convergence rates for regular and singular solutions with the (adaptive) h-version and (augmented) p-version.

In this paper, we present a meshless discretization technique for instationary convection-di usion problems. It is based on operator splitting, the method of characteristics and a generalized partition of unity method. We focus on the discretization process and its quality. The method may be used as an h-or p-version. Even for general particle distributions, the convergence behavior of the di erent versions corresponds to that of the respective version of the nite element method on a uniform grid. We discuss the implementational aspects of the proposed method. Furthermore, we present the results of numerical examples, where we considered instationary convection-di usion, instationary di usion, linear advection and elliptic problems.

Abstract. We present an algebraic multigrid (AMG) method for the efficient solution of linear (block-)systems stemming from a discretization of a system of partial differential equations (PDEs). It generalizes the classical AMG approach for scalar problems to systems of PDEs in a natural blockwise fashion. We apply this approach to linear elasticity and show that the block-interpolation, described in this paper, reproduces the rigid body modes, i.e., the kernel elements of the discrete linear elasticity operator. It is well-known from geometric multigrid methods that this reproduction of the kernel elements is an essential property to obtain convergence rates which are independent of the problem size. We furthermore present results of various numerical experiments in two and three dimensions. They confirm that the method is robust with respect to variations of the Poisson ratio ν. We obtain rates ρ < 0.4 for ν < 0.4. These measured rates clearly show that the method provides fast convergence for a large variety of discretized elasticity problems.

Abstract. In this sequel to [15,16] we focus on the efficient solution of the linear block-systems arising from a Galerkin discretization of an elliptic partial differential equation of second order with the partition of unity method (PUM). We present a cheap multilevel solver for partition of unity discretizations of any order. The shape functions of a PUM are products of piecewise rational partition of unity (PU) functions ϕ i with supp(ϕ i ) = ω i and higher order local approximation functions ψ n i (usually a local polynomial of degree ≤ p i ). Furthermore, they are non-interpolatory. In a multilevel approach we not only have to cope with non-interpolatory basis functions but also with a sequence of nonnested spaces due to the meshfree construction. Hence, injection or interpolatory interlevel transfer operators are not available for our multilevel PUM. Therefore, the remaining natural choice for the prolongation operators are L 2 -projections. Here, we exploit the partition of unity construction of the function spaces and a hierarchical construction of the PU itself to localize the corresponding projection problem. This significantly reduces the computational costs associated with the setup and the application of the interlevel transfer operators. The second main ingredient for our multilevel solver is the use of a block-smoother to treat the local approximation functions ψ n i for all n simultaneously. The results of our numerical experiments in two and three dimensions show that the convergence rate of the proposed multilevel solver is independent of the number of patches card({ω i }). The convergence rate is slightly dependent on the local approximation orders p i .

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.