On the numerical solution of the first biharmonic equation

Peisker, P.

doi:10.1051/m2an/1988220406551

Cited by 29 publications

(20 citation statements)

References 12 publications

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“…Higher order accuracy can be achieved using finite element methods. There are many finite element approaches which use iterative methods, such as those in [6,9,10,18,23]. Moreover, Hermite bicubic orthogonal spline collocation (OSC) methods for the biharmonic Dirichlet problem (1.1) have been considered by Cooper and Prenter [11] who proposed an alternating direction implicit OSC method, and by Sun [27] who presented a Schur complement OSC algorithm, the cost of which is O(N 3 log 2 N ).…”

Section: Introductionmentioning

confidence: 99%

Orthogonal spline collocation methods for biharmonic problems

Lou

Bialecki

Fairweather

1998

Numerische Mathematik

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Section: Introductionmentioning

confidence: 99%

Orthogonal spline collocation methods for biharmonic problems

Lou

Bialecki

Fairweather

1998

Numerische Mathematik

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“…This can be derived by choosing = 1/(2 + 2 2 ). Replacing * * by max{2 + 1 , 2 2 + 1} will lead to a more tractable expression for the optimal parameter opt which minimizes the condition number in (32), when a scaling parameter >0 is introduced in the augmented Lagrangian formulation and A …”

Section: Remark 11mentioning

confidence: 99%

Analysis of block matrix preconditioners for elliptic optimal control problems

Mathew

Sarkis

Schaerer

2007

Numerical Linear Algebra App

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SUMMARYIn this paper, we describe and analyse several block matrix iterative algorithms for solving a saddle point linear system arising from the discretization of a linear-quadratic elliptic control problem with Neumann boundary conditions. To ensure that the problem is well posed, a regularization term with a parameter is included. The first algorithm reduces the saddle point system to a symmetric positive definite Schur complement system for the control variable and employs conjugate gradient (CG) acceleration, however, double iteration is required (except in special cases). A preconditioner yielding a rate of convergence independent of the mesh size h is described for ⊂ R 2 or R 3 , and a preconditioner independent of h and when ⊂ R 2 . Next, two algorithms avoiding double iteration are described using an augmented Lagrangian formulation. One of these algorithms solves the augmented saddle point system employing MINRES acceleration, while the other solves a symmetric positive definite reformulation of the augmented saddle point system employing CG acceleration. For both algorithms, a symmetric positive definite preconditioner is described yielding a rate of convergence independent of h. In addition to the above algorithms, two heuristic algorithms are described, one a projected CG algorithm, and the other an indefinite block matrix preconditioner employing GMRES acceleration. Rigorous convergence results, however, are not known for the heuristic algorithms.

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“…2i , (4.41) where Following the idea used in [11] for C 0 piecewise linear finite elements, the preconditioner for S is defined by…”

Section: Algorithm IImentioning

confidence: 99%

“…The presence of R in P is motivated by the fact (see Proposition 7.2 and (7.6) in [11]) that for C 0 piecewise linear finite elements on [0, 1] vanishing at 0 and 1, their Sobolev norm · H −1/2 (0,1) is equivalent to the discrete energy norm generated by the matrix R. The constant 1/4 in P was selected experimentally to minimize κ 2 (P −1/2 SP −1/2 ). Numerical tests in Table 1 indicate that κ 2 (P −1/2 SP −1/2 ), equal to λ max (P −1 S)/λ min (P −1 S), is bounded from above by a small constant which is independent of N. It is well-known (see, for example, Section 3 in [11]) that, with the use of FFTs, the cost of solving a system with R is O (N log 2 N).…”

Section: Algorithm IImentioning

confidence: 99%

“…In the mixed approach of [6], (1.1) is first reduced to a coupled system of two second order partial differential equations in u and v = u. The coupled system in u and v is then discretized using, for example, the five-point finite difference formula [7], or the nine-point finite difference formula [2], or C 0 piecewise linear finite elements [11], or C 1 Hermite cubic finite elements [9]. Some of finite difference and finite element methods are very efficient.…”

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confidence: 99%

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Modified nodal cubic spline collocation for biharmonic equations

Abushama

Bialecki

2007

Numer Algor

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We formulate a modified nodal cubic spline collocation scheme for the solution of the biharmonic Dirichlet problem on the unit square. We prove existence and uniqueness of a solution of the scheme and show how the scheme can be solved on an N × N uniform partition of the square at a cost O(N 2 log 2 N + mN 2 ) using fast Fourier transforms and m iterations of the preconditioned conjugate gradient method. We demonstrate numerically that m proportional to log 2 N guarantees the desired convergence rates. Numerical results indicate the fourth order accuracy of the approximations in the global maximum norm and the fourth order accuracy of the approximations to the first order partial derivatives at the partition nodes.

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On the numerical solution of the first biharmonic equation

Cited by 29 publications

References 12 publications

Orthogonal spline collocation methods for biharmonic problems

Orthogonal spline collocation methods for biharmonic problems

Analysis of block matrix preconditioners for elliptic optimal control problems

Modified nodal cubic spline collocation for biharmonic equations

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