FIESTA: THE p-VERSION APPROACH IN FINITE ELEMENT ANALYSIS

Block SSOR preconditionings for high order 3d fe systems

“…The 3D Navier equations are very important for structural analysis and are frequently considered in connection with p-adaptive versions of FEM (see, e.g., [2,3,4,25]). This model problem is approximated using hierarchical finite elements of order p, 2 <~ p ~< 5, with shape functions of the form ~i(x)~j(y)tp~(z), where co(t) = 0.5(1 -t), ~01(t ) = 0.5(1 + t), ~l(t) = Pt(t) -Pl 2(0, l ~> 2, Pl(t) is the Legendre polynomial of degree l. To study properties of the related sparse SPD matrices and the efficiency of the suggested numerical methods for solving the corresponding linear systems we consider hierarchical FE approximations of problem (2.1) on a uniform, a slightly nonuniform and a highly nonuniform 8 x 8 x 8 element mesh (Test Problems I, 2 and 3, respectively) each with three values of the Poisson ratio: v = 0.45, 0.47, and 0.49.…”

“…This agreement probably gives evidence for a lack of noticeable eigenvalue clustering. The k n o w n programs S T R I P E [2,3], F I E S T A [4] and P R O B E [25] implementing p-adaptive versions of F E M for 3D problems solve the resulting linear algebraic systems by the sparse Cholesky factorization m e t h o d which requires excessive m e m o r y to store the matrix triangular factors and e n o r m o u s arithmetic costs. F o r instance, for a mesh of 189 hierarchical finite elements of degree p = 6 the Cholesky decomposition of the resulting coefficient matrix using highly vectorized code needs 3800s of the CRAY-1 CPU time/-2, 3].…”