L. F. Pavarino. Domain Decomposition Algorithms for the p-version Finite Element Method for Elliptic Problems. PhD thesis, Courant Institute of Mathematical Sciences, New York University, September 1992. also Technical Report, Nr. 616, Department of Computer Science, Courant Institute.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....discretizations of these equations; see, e.g. 67, 68, 98] For both families of discretizations, the design of preconditioners for three dimensional problems is especially challenging. Early work on preconditioners for spectral methods was carried out by Canuto and Funaro [35] and Pavarino [91, 92, 93]. Some of the algorithms studied by Pavarino are numerically scalable (i.e. the number of iterations is independent of the number of substructures) and optimal (the number of iterations does not grow or grows only slowly with the degree of the polynomials) However, each application 24 of his ....

....see e.g. 65] We also note that when the overlap is generous, the method is optimal in the sense that the condition number is uniformly bounded with respect to N and H. Remark 3.5. 2 In the present algorithm, the local spaces are allowed to be more general than those considered by Pavarino [91, 92, 93]. For each crosspoint x , Pavarino defines an extended subdomain Omega 0 as the union of all the subdomains that contain x as a vertex. Therefore, ffi is always on the order of H. We now apply the FEM SEM equivalence to the subspaces that define K h;AS , to propose yet another ....

Luca F. Pavarino. Domain Decomposition Algorithms for the p-version Finite Element Method for Elliptic Problems. PhD thesis, Courant Institute, New York University, September 1992.

....element discretizations of these equations; see e.g. 16, 17, 29] For both families of discretizations, the design of preconditioners for three dimensional problems is especially challenging. Early work on preconditioners for spectral methods was carried out by Canuto and Funaro [7] and Pavarino [24, 25, 26]. Some of the algorithms studied by Pavarino are numerically scalable (i.e. the number of iterations is independent of the number of substructures) and optimal (the number of iterations does not grow or grows only slowly with the degree of the polynomials) However, each application of his ....

....see e.g. 15] We also note that when the overlap is generous, the method is optimal in the sense that the condition number is uniformly bounded with respect to N and H. Remark 2. In the present algorithm, the local spaces are allowed to be more general than those considered by Pavarino [24, 25, 26]. For each crosspoint x , Pavarino defines an extended subdomain Omega 0 as the union of all the subdomains that contain x as a vertex. Therefore, ffi is always on the order of H. We now apply the FEM SEM equivalence to the subspaces that define K h;AS , to propose yet another ....

L. F. Pavarino, Domain Decomposition Algorithms for the p-version Finite Element Method for Elliptic Problems, PhD thesis, Courant Institute, New York University, September 1992.

.... long range interactions of the basis elements produce quite dense and expensive factorizations of the stiffness matrix, and the use of direct methods is not economical due to the large memory requirements [12] Early work on preconditioners for these equations was done by Pavarino [20] 21] [19]. His algorithms are numerically scalable (i.e. the number of iterations is independent of the number of substructures) and quasi optimal (the number of iterations grows slowly with the degree of the polynomials. However, each application of the preconditioner can be very expensive. Several ....

....when the coefficient k has large jumps, such a growth is very moderate in numerical experiments; see e. g [13] We note also that when the overlap is generous, the method is optimal in the sense that the condition number is uniformly bounded with respect to the parameters of the problem; see [19] for early work on this type of preconditioner. Our results and techniques allow a very flexible choice of subregions. We now apply FEM SEM equivalence to the subspaces used to define B h;AS ; this is the same technique used to derive the preconditioner SN;WB from S h;WB . The coarse space is the ....

Luca F. Pavarino. Domain Decomposition Algorithms for the p-version Finite Element Method for Elliptic Problems. PhD thesis, Courant Institute, New York University, September 1992.

....For all these methods, improved accuracy is achieved by increasing the spectral degree aas well as the number of elements. We note that iterative solvers for a variety of higher order methods have been developed by Mandel [24] Katz and Hu [18] and Guo [16] see also the theses of Pavarino [28], Casarin [9] and Bica [4] In our previous work [29, 31] we considered the scalar case and iterative substructuring methods with a wire basket based coarse space. Each spectral element is then the affine image of a reference cube considered as a subdomain of the domain decomposition method. The ....

L. F. Pavarino, Domain Decomposition Algorithms for the p-version Finite Element Method for Elliptic Problems, PhD thesis, Dept. of Mathematics, Courant Institute of Mathematical Sciences, New York University, September 1992.

....the degree p of the polynomial basis functions is increased only in selected subregions. Optimal and almost optimal bounds are obtained, depending on the choice of refinement subdomains. Numerical experiments confirming these results, as well as proofs and details, can be found in Pavarino [14] [13]. 2. The Model Problem. We consider a model problem for linear, self adjoint, second order elliptic problems, on a bounded Lipschitz region Omega Gamma Dirichlet boundary conditions are given on Gamma D , a closed subset of Omega with positive measure, and Neumann conditions are given on ....

....point of Omega r . This result specifies which choices of refinement points lead to a bounded condition number. It is interesting to note that if a whole edge is isolated on Omega r , then we will still have a constant bound. The proof of this theorem can be found in Pavarino [13] and is based on a series of technical results concerning the decomposition of discrete harmonic polynomials and on Theorem 3.1. The main tools used in the proof are Markov s theorem (see Rivlin [15] and a p version analog of the decomposition lemma 3.2 in Widlund [17] A two dimensional ....

L. F. Pavarino, Domain Decomposition Algorithms for the p-version finite element method for elliptic problems, PhD thesis, Courant Institute of Mathematical Sciences, September 1992. In preparation.

....here is directly inspired by a method developed by Barry Smith [32,33] for the h version. Important progress has previously been reported, for problems in two dimensions, in Babuska, Craig, Mandel, and Pitkaranta [1] in which polylogarithmic bounds for some methods are proved; see also Pavarino [27] for results in two dimensions, which are similar to those of this paper. In three dimensions, pioneering work has been carried out by Jan Mandel [25,24,23,26] His algorithms, which use global spaces which differ from ours, have also been implemented in industrial software. A number of domain ....

....Bernardi and Maday [4] However, we know of no previous theoretical results that show only polynomial growth, in log p; for problems in three dimensions. We note that other domain decomposition algorithms for higher order methods, based on overlapping subregions, have been considered in Pavarino [28,27]. We note that it is known that certain collocation methods result in coefficient matrices which are spectrally equivalent to the stiffness matrices derived from the Galerkin procedure considered here; cf. Bernardi and Maday [4] It therefore appears likely that our algorithm could be of use ....

Luca F. Pavarino. Domain Decomposition Algorithms for the p-version Finite Element Method for Elliptic Problems. PhD thesis, Courant Institute, New York University, September 1992.

....they differ from the Schwarz methods that use overlapping subregions; see, e.g. Dryja and Widlund [23, 24] for a discussion of recent work on this other major family of methods. We also note that similar results, for higher order methods and both two and three dimensions, are given in Pavarino [43, 42]. All these iterative methods are thus two level methods and convincing arguments have been put forward supporting the opinion that they are particularly well suited for the large, relatively loosely coupled computing systems that are becoming increasingly common; cf. Gropp [29] The best of these ....

.... basic result has previously been announced in Pavarino [44] and Pavarino and Widlund [45] Important progress has previously been reported, for problems in two dimensions, by Babuska, Craig, Mandel, and Pitkaranta [1] in which polylogarithmic bounds for some methods are proved; see also Pavarino [42] for results in two dimensions, which are similar to those of this paper. Early experimental work is reported in Babuska and Elman [2] In three dimensions, pioneering work has been carried out by Mandel [38, 37, 36, 41] Some of his algorithms, which use global spaces which differ from ours, are ....

Luca F. Pavarino. Domain Decomposition Algorithms for the p-version Finite Element Method for Elliptic Problems. PhD thesis, Courant Institute, New York University, September 1992.

....of iterative substructuring methods is more challenging, since the stiffness matrices can be much more ill conditioned and different mathematical tools are needed. See Babuska, Craig, Mandel, and Pitkaranta [1] for two dimensional problems, Mandel [9, 10] for three dimensional problems, Pavarino [12, 11] for overlapping methods in two and three dimensions, R nquist [15] and Maday and Patera [7] for spectral element methods. In this paper, we propose a wire basket based algorithm (in the terminology of Dryja, Smith, and Widlund [4] with condition number bounded by (1 log p) 2 , where p is the ....

Luca F. Pavarino. Domain Decomposition Algorithms for the p-version Finite Element Method for Elliptic Problems. PhD thesis, Courant Institute of Mathematical Sciences, September 1992. Technical Report 616.

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L. F. Pavarino. Domain Decomposition Algorithms for the p-version Finite Element Method for Elliptic Problems. PhD thesis, Courant Institute of Mathematical Sciences, New York University, September 1992. also Technical Report, Nr. 616, Department of Computer Science, Courant Institute.

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