Efficient implementation of domain decomposition methods using a hierarchical h-adaptive finite element program

A previous contribution[

1

] showed the hierarchical relationships between parent and child elements that come out if these elements are geometrically similar. Under this similarity condition, the terms involved in the evaluation of element stiffness matrices (

k

e

=∫

B

t

DB

∣

J

∣d

V

), corresponding to parent and child elements, are related by a constant which is a function of the ratio of the element sizes (scaling factor). These relations were used in the basic implementation of a hierarchical

h

-adaptive Finite Element program based on element subdivision for the resolution of the 2-D linear elasticity problem. The program makes use of a hierarchical data structure to carry out the

h

-adaptive process, which significantly reduces the amount of calculations required for the evaluation of the problem stiffness matrix, element stresses, element error estimation,...

The

h

-adaptive refinement process based on element splitting produces a natural decomposition of the domain which, together with the hierarchical data structure of the program directly produces an

arrowhead

stiffness matrix allowing for a decomposition of the global problem into smaller problems. Thus, a domain decomposition solver has been used in this paper to efficiently solve the linear system of equations arising during the analysis process. The numerical test presented in the paper clearly show a considerable improvement in memory requirements and solution times and suggest the use of recursive domain decomposition into the original subdomains.