Multilevel algorithms for Rannacher–Turek finite element approximation of 3D elliptic problems

• Published in: Computing Vol. 82; no. 4; pp. 217 - 239
• Main Authors: Georgiev, I., Kraus, J., Margenov, S.
• Format: Journal Article
• Language: English
• Published: Vienna Springer Vienna 01.09.2008; Springer; Springer Nature B.V
• Subjects: Algorithmics. Computability. Computer arithmetics; Algorithms; Applied mathematics; Applied sciences; Approximation; Artificial Intelligence; Computer Appl. in Administrative Data Processing; Computer Communication Networks; Computer Science; Computer science; control theory; systems; Exact sciences and technology; Finite element analysis; Information Systems Applications (incl.Internet); Mathematics; Methods of scientific computing (including symbolic computation, algebraic computation); Numerical analysis; Numerical analysis. Scientific computation; Sciences and techniques of general use; Software Engineering; Studies; Theoretical computing; Multilevel preconditioners; Non-conforming FEM; CBS constant; Hierarchical basis; 65N22; 65N55; 65N30; Freedom degree; Cauchy Schwarz inequality; Approximation; Three-dimensional calculations; Grid pattern; Non conforming finite element; Iterative method; Non-conforming FEM . Multilevel preconditioners . Hierarchical basis . CBS constant . 65N30; Experimental study; Algorithm; Computational complexity; Partial differential equation; Convergence; Finite element method; Splitting; Numerical analysis; Elliptic equation; Scientific computation; Conforming finite element; Two-dimensional calculations
• ISSN: 0010-485X; 1436-5057
• DOI: 10.1007/s00607-008-0008-5

Generalizing the approach of a previous work of the authors, dealing with two-dimensional (2D) problems, we present multilevel preconditioners for three-dimensional (3D) elliptic problems discretized by a family of Rannacher Turek non-conforming finite elements. Preconditioners based on various multilevel extensions of two-level finite element methods (FEM) lead to iterative methods which often have an optimal order computational complexity with respect to the number of degrees of freedom of the system. Such methods were first presented by Axelsson and Vassilevski in the late-1980s, and are based on (recursive) two-level splittings of the finite element space. An important point to make is that in the case of non-conforming elements the finite element spaces corresponding to two successive levels of mesh refinement are not nested in general. To handle this, a proper two-level basis is required to enable us to fit the general framework for the construction of two-level preconditioners for conforming finite elements and to generalize the method to the multilevel case. In the present paper new estimates of the constant γ in the strengthened Cauchy–Bunyakowski–Schwarz (CBS) inequality are derived that allow an efficient multilevel extension of the related two-level preconditioners. Representative numerical tests well illustrate the optimal complexity of the resulting iterative solver, also for the case of non-smooth coefficients. The second important achievement concerns the experimental study of AMLI solvers applied to the case of micro finite element ( μ FEM) simulation. Here the coefficient jumps are resolved on the finest mesh only and therefore the classical CBS inequality based convergence theory is not directly applicable. The obtained results, however, demonstrate the efficiency of the proposed algorithms in this case also, as is illustrated by an example of microstructure analysis of bones.