Particle–partition of Unity Methods in Elasticity

The particle–partition of unity method (PUM) [1, 2, 3, 4, 5, 8] is a meshfree Galerkin method for the numerical treatment of partial differential equations (PDE). In essence, it is a generalized finite element method (GFEM) which employs piecewise rational shape functions rather than piecewise polynomial functions. The PUM shape functions, however, make up a basis of the discrete function space unlike other GFEM approaches which allows us to construct fast multilevel solvers in a similar fashion as in the finite element method (FEM). The paper is organized as follows: In section 2 we shortly review the construction of PUM spaces, the Galerkin discretization of a linear elliptic PDE using our PUM as well as the fast multilevel solution of the arising linear system. Then we present some numerical results with respect to approximation as well as fast solution techniques in two and three space dimensions obtained with our PUM for the numerical solution of the Navier–Lamé equations in section 2.4. The discretization of constrained minimization problems like the obstacle problem is the subject of section 3. Then, some numerical results for the obstacle problem in two space dimensions are given in section 3.2. Finally, we conclude with some remarks.