AN OVERLAPPING SCHWARZ ALGORITHM FOR ALMOST INCOMPRESSIBLE ELASTICITY

• Published in: SIAM journal on numerical analysis Vol. 47; no. 4; pp. 2897 - 2923
• Main Authors: DOHRMANN, CLARK R., WIDLUND, OLOF B.
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2009
• Subjects: Algorithms; Aspect ratio; Boundary conditions; Coarsening; Cubes; Degrees of freedom; Elasticity; Finite element method; Harmonic functions; Laboratories; Mathematical analysis; Methods; Numerical analysis; Poisson ratio; Poissons ratio; Rigid structures; Tetrahedrons; Vertices
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/080724320

Overlapping Schwarz methods are extended to mixed finite element approximations of linear elasticity which use discontinuous pressure spaces. The coarse component of the preconditioner is based on a low-dimensional space previously developed for scalar elliptic problems and a domain decomposition method of iterative substructuring type, i.e., a method based on nonoverlapping decompositions of the domain, while the local components of the preconditioner are based on solvers on a set of overlapping subdomains. A bound is established for the condition number of the algorithm which grows in proportion to the logarithm of the number of degrees of freedom in individual subdomains and, essentially, to the third power of the relative overlap between the overlapping subdomains, and which is independent of the Poisson ratio as well as jumps in the Lamé parameters across the interface between the subdomains. A positive definite reformulation of the discrete problem makes the use of the standard preconditioned conjugate gradient method straight-forward. Numerical results, which include a comparison with problems of compressible elasticity, illustrate the findings.