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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Numerische Mathematik 73 (1996), S. 1-36 
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):65F10, 65N20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. We develop and analyze a procedure for creating a hierarchical basis of continuous piecewise linear polynomials on an arbitrary, unstructured, nonuniform triangular mesh. Using these hierarchical basis functions, we are able to define and analyze corresponding iterative methods for solving the linear systems arising from finite element discretizations of elliptic partial differential equations. We show that such iterative methods perform as well as those developed for the usual case of structured, locally refined meshes. In particular, we show that the generalized condition numbers for such iterative methods are of order $J^2$ , where $J$ is the number of hierarchical basis levels.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 41 (1998), S. 499-526 
    ISSN: 0029-5981
    Keywords: multigrid method ; finite element discretization ; matrix consistency ; grid generation ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Increasing the efficiency of solving linear/linearized matrix equations is a key point to save computer time in numerical simulation, especially for three-dimensional problems. The multigrid method has been determined to be efficient in solving boundary-value problems. However, this method is mostly linked to the finite difference discretization, rather than to the finite element discretization. This is because the grid relationship between fine and coarse grids was not achieved effectively for the latter case. Consequently, not only is the coding complicated but also the performance is not satisfactory when incorporating the multigrid method into the finite element discretization. Here we present an approach to systematically prepare necessary information to relate fine and coarse grids regarding the three-dimensional finite element discretization, such that we can take advantage of using the multigrid method. To achieve a consistent approximation at each grid, we use A2h=I2hhAhIh2h and b2h=I2hh bh, starting from the composed matrix equation of the finest grid, to prepare the matrix equations for coarse grids. Such a process is implemented on an element level to reduce the computation to its minimum. To demonstrate the performance, this approach has been used to adapt two existing three-dimensional finite element subsurface flow and transport models, 3DFEMWATER and 3DLEWASTE, to their multigrid version, 3DMGWATER and 3DMGWASTE, respectively. Two example problems, one for each model, are considered for illustration. The computational result shows that the multigrid method can help solve the example problems very efficiently with our presented modular setting. © 1998 John Wiley & Sons, Ltd.
    Additional Material: 11 Ill.
    Type of Medium: Electronic Resource
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  • 3
    Title: Domain decomposition methods in scientific and engineering computing : proceedings of the 7th international conference on domain decomposition methods, 1993, Pennsylvania State University; 180
    Contributer: Keyes, David E. , Xu, Jinchao
    Publisher: Providence, RI :AMS,
    Year of publication: 1994
    Pages: 546 S.
    Series Statement: Contemporary mathematics 180
    Type of Medium: Book
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