Triangular Iterative Methods

Samarskiî, A. A.; Nikolaev, Evgenii S.

doi:10.1007/978-3-0348-9142-4_5

Cited by 18 publications

(24 citation statements)

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“…By selecting the relevant rows of (19), (20), and (21), one may finally construct a new PSO for the combined mesh block. One may discard the rows of (19), (20), and (21) corresponding to nodes that do not belong to edge elements of the merged block; these rows (equations) are totally extraneous to the PSO of the merged block of mesh.…”

Section: Merging Two Psosmentioning

confidence: 99%

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Fast finite element calculation of effective conductivity of random continuum microstructures: The recursive Poincaré–Steklov operator method

Hardin

Schuh

2017

Journal of Computational Physics

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The effective conductivity of a block of composite can be extracted from the Dirichletto-Neumann Poincaré-Steklov operator (PSO) for that block. In this paper, a domain decomposition method for computing the PSO over a finite element mesh is discussed. A new numerical strategy is introduced to accelerate the computation of this operator, using the Schur complement to calculate the PSO for the smallest subdomains, then recursively merging subdomain PSOs up to the full domain. At each step of the algorithm, information extraneous to the PSO is discarded. The effective conductivity values computed by this method are identical to those obtained from a basic Finite Element Method, an order of magnitude faster and with much less computer memory consumed. As proof of concept, effective conductivity measurements are presented for a percolating random fractal-like microstructure across a range of phase fractions.

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Section: Merging Two Psosmentioning

confidence: 99%

“…(2). This method is extremely flexible and well-studied [14][15][16][17][18][19][20], with wide application in solving partial differential equations of many kinds. Using the finite element method to solve Eqn.…”

Section: Introductionmentioning

confidence: 99%

Fast finite element calculation of effective conductivity of random continuum microstructures: The recursive Poincaré–Steklov operator method

Hardin

Schuh

2017

Journal of Computational Physics

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“…with coefficients which depend on the parameters σ 0 , · · · , σ mi , α 0 , · · · , α mi−1 given in the definition of the CCG algorithm (see [10]). Here and in the following we denote by I the identity in the corresponding space.…”

Section: The Cascadic Algorithmmentioning

confidence: 99%

“…Here and in the following we denote by I the identity in the corresponding space. From [10] we recall the wellknown optimality property of the CG-algorithm: …”

Section: The Cascadic Algorithmmentioning

confidence: 99%

The Convergence of the Cascadic Conjugate-Gradient Method under a Deficient Regularity

Shaidurov

1994

Problems and Methods in Mathematical Physics

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Abstract. We study the convergence properties of the cascadic conjugategradient method (CCG-method), which can be considered as a multilevel method without coarse-grid correction. Nevertheless, the CCG-method converges with a rate that is independent of the number of unknowns and the number of grid levels. We prove this property for two-dimensional elliptic second-order Dirichlet problems in a polygonal domain with an interior angle greater than π. For piecewise linear finite elements we construct special nested triangulations that satisfy the conditions of a "triangulation of type (h, γ, L)" in the sense of I. Babuška, R. B. Kellogg and J. Pitkäranta. In this way we can guarantee both the same order of accuracy in the energy norm of the discrete solution and the same convergence rate of the CCG-method as in the case of quasiuniform triangulations of a convex polygonal domain.

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“…We obtain H 1 and H 2 error estimates using a generalization of Banach's theorem [13]. The quadratic convergence of Newton's method for the solution of the OSC problem is proved in a way similar to that in [22].…”

Section: Introductionmentioning

confidence: 99%

Orthogonal Spline Collocation for Nonlinear Dirichlet Problems

Aitbayev

Bialecki²

2000

SIAM J. Numer. Anal.

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Abstract. We study the orthogonal spline collocation (OSC) solution of a homogeneous Dirichlet boundary value problem in a rectangle for a general nonlinear elliptic partial differential equation. The approximate solution is sought in the space of Hermite bicubic splines. We prove local existence and uniqueness of the OSC solution, obtain optimal order H 1 and H 2 error estimates, and prove the quadratic convergence of Newton's method for solving the OSC problem.

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Triangular Iterative Methods

Cited by 18 publications

References 0 publications

Fast finite element calculation of effective conductivity of random continuum microstructures: The recursive Poincaré–Steklov operator method

Fast finite element calculation of effective conductivity of random continuum microstructures: The recursive Poincaré–Steklov operator method

The Convergence of the Cascadic Conjugate-Gradient Method under a Deficient Regularity

Orthogonal Spline Collocation for Nonlinear Dirichlet Problems

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