Preconditioning of the p-version of the finite element method

On Fast Domain Decomposition Solving Procedures for hp-Discretizations of 3-D Elliptic Problems

“…In particular, at the indicated out special ordering of unknowns, we have A I = diag [A e,e , A o,e , A e,o , A o,o ] and M I = diag[M e,e , M o,e , M e,o , M o,o ] with the blocks defined similarly to (4.2), but with differently defined diagonal and tridiagonal matrices, participating in Kronecker's products. For the explicit expressions see [16,25].…”

“…For the discretizations of 2-d elliptic problems, all basic components of DD solvers have been reasonably well studied and made indeed efficient. Due to [1,12,16,23,25,26] a number of Schur complement preconditioners has been designed, optimal or almost optimal in condition and requiring less, sometimes considerably, than O(p The same as for the reference element and with the same total arithmetical costs DD and multigrid like preconditioners-solvers are applicable to local Dirichlet problems on subdomains of decomposition -otherwise the most time consuming subproblems in DD algorithms. In order to be true, this requires only the conditions of the generalized angular quasiuniformity to be fulfilled.…”

“…It was derived as a wavelet multiresolution preconditioner for the matrix, obtained from the finite-difference like preconditioner of [25] by neglecting the minor term at the cost of (1 + log p)…”

“…If the mappings are trilinear, i.e., elements have straight edges, these conditions are equivalent to shape regularity. In a more general case, they imply that the inequalities in the first line of (4.3) in [25] are fulfilled with h = h r specific for each element, but with α (1) and θ independent of an element and h r . In this paper, we consider hp discretizations with the hierarchical reference element, which is equipped with the space Q p,x of all polynomials of the order not greater than p , p 1, in each variable.…”

“…This assumption is typical of many other works on fast solvers for hp discretizations, e.g., see [1,4,5,7,8,12,[15][16][17][18][19][20][21][22][23][24][25][26][27][28]31,32]. The setup component of the solver greately depends on the computer representation of the boundary of the domain and on the related mappings X (r) (y), which may be sub-, super-or isoparametric, or may be given by some analytical functions, exactly representing the boundary, e.g., as in [21,22].…”