The Numerical Solution of the Biharmonic Equation, Using a Spectral Multigrid Method

Vries, R. M. de; Zandbergen, P.J.

doi:10.1007/978-1-4612-3684-9_3

Cited by 7 publications

(8 citation statements)

References 8 publications

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“…This problem has been studied by a number of authors (see, e.g., Kelmanson [12] and de Vries and Zandbergen [17]). Kelmanson solved this problem by employing a modified integral equation.…”

Section: Convergence Ratesmentioning

confidence: 99%

See 1 more Smart Citation

Multigrid Solution of Automatically Generated High-Order Discretizations for the Biharmonic Equation

Altas¹,

Dym²,

Gupta³

et al. 1998

SIAM J. Sci. Comput.

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In this work, we use a symbolic algebra package to derive a family of finite difference approximations for the biharmonic equation on a 9-point compact stencil. The solution and its first derivatives are carried as unknowns at the grid points. Dirichlet boundary conditions are thus incorporated naturally. Since the approximations use the 9-point compact stencil, no special formulas are needed near the boundaries. Both second-order and fourth-order discretizations are derived.The fourth-order approximations produce more accurate results than the 13-point classical stencil or the commonly used system of two second-order equations coupled with the boundary condition.The method suffers from slow convergence when classical iteration methods such as Gauss-Seidel or SOR are employed. In order to alleviate this problem we propose several multigrid techniques that exhibit grid-independent convergence and solve the biharmonic equation in a small amount of computer time. Test results from three different problems, including Stokes flow in a driven cavity, are reported.

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“…This problem has been studied by a number of authors (see, e.g., Kelmanson [12] and de Vries and Zandbergen [17]). Kelmanson solved this problem by employing a modified integral equation.…”

Section: Convergence Ratesmentioning

confidence: 99%

“…The coupled equation approach has been used by many authors (see [8], [9], and other references for detailed background). There also have been efforts to introduce multigrid techniques with the coupled equation approach [3], [14], [15], [17].…”

Section: Introduction Consider the Dirichlet Problem For The Biharmomentioning

confidence: 99%

Multigrid Solution of Automatically Generated High-Order Discretizations for the Biharmonic Equation

Altas¹,

Dym²,

Gupta³

et al. 1998

SIAM J. Sci. Comput.

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“…The multiple integrals occurring in the problem can be reduced to simple summations of the Chebyshev coefficients. As an efficient and accurate numerical method for solving the Poisson problem for the streamfunction, a spectral multigrid method is used, see [12,13].…”

Section: Numerical Resultsmentioning

confidence: 99%

“…After substitution of the extremizer 05(2) into (12), the optimization for the multiplier(s) becomes finite-dimensional and the optimization for the function H(05)-2(1(05)-7) can be solved by a steepest descent method. The numerical optimization for to is infinite-dimensional however, and moreover, the functional H(o))-2(1(o))-7) might be nonconvex in ~.…”

Section: Variational Formulation For Lagrange Multipliersmentioning

confidence: 99%

Numerical algorithm for the calculation of nonsymmetric dipolar and rotating monopolar vortex structures

Fliert

Roo

Vries

et al. 1995

Journal of Computational and Applied Mathematics

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A numerical iteration scheme is presented for the calculation of coherent vortex structures. Steady solutions of the Euler vorticity equation are found, using a variational characterization for dipolar and monopolar vortices as relative equilibria of the Poisson system. The variational principle for the vorticity is solved by a numerical method for nonconvex optimization. Besides the variational principle for the vorticity, an optimization process is used for the multipliers that appear in the description. The free boundary is solved implicitly in the iteration process.

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“…The main difficulty with this approach is that the boundary conditions for the new variable v are undefined and need to be approximated from the discrete form of eu ¼ v. The coupled equation approach has been used by Gupta (1975). There have also been considerable efforts to introduce multigrid methods for the coupled equation approach (Altas et al, 2002;Devries and Zandbergen, 1989). Another approach for solving the 2D biharmonic equations is to discretise the differential equation on a uniform grid using a 13-point approximation with truncation error of order h 2 or using a wider computational stencil such as the 25-point approximation with truncation error of order h 4 .…”

Section: Introductionmentioning

confidence: 99%

Solution of the two dimensional second biharmonic equation with high‐order accuracy

Dehghan

Mohebbi

2008

Kybernetes

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If you would like to write for this, or any other Emerald publication, then please use our Emerald for Authors service information about how to choose which publication to write for and submission guidelines are available for all. Please visit www.emeraldinsight.com/authors for more information. About Emerald www.emeraldinsight.comEmerald is a global publisher linking research and practice to the benefit of society. The company manages a portfolio of more than 290 journals and over 2,350 books and book series volumes, as well as providing an extensive range of online products and additional customer resources and services.Emerald is both COUNTER 4 and TRANSFER compliant. The organization is a partner of the Committee on Publication Ethics (COPE) and also works with Portico and the LOCKSS initiative for digital archive preservation. AbstractPurpose -The purpose of this paper is to introduce efficient methods for solving the 2D biharmonic equation with Dirichlet boundary conditions of second kind. This equation describes the deflection of loaded plate with boundary conditions of simply supported plate kind. Also it can be derived from the calculus of variations combined with the variational principle of minimum potential energy. Because of existing fourth derivatives in this equation, introducing high-order accurate methods need to use artificial points. Also solving the resulted linear system of equations suffers from slow convergence when iterative methods are used. This paper aims to introduce efficient methods to overcome these problems. Design/methodology/approach -The paper considers several compact finite difference approximations that are derived on a nine-point compact stencil using the values of the solution and its second derivatives as the unknowns. In these approximations there is no need to define special formulas near the boundaries and boundary conditions can be incorporated with these techniques. Several iterative linear systems solvers such as Krylov subspace and multigrid methods and their combination (with suitable preconditioner) have been developed to compare the efficiency of each method and to design powerful solvers. Findings -The paper finds that the combination of compact finite difference schemes with multigrid method and Krylov iteration methods preconditioned by multigrid have excellent results for the second biharmonic equation, and that Krylov iteration methods preconditioned by multigrid are the most efficient methods. Originality/value -The paper is of value in presenting, via some tables and figures, some numerical experiments which resulted from applying new methods on several test problems, and making comparison with conventional methods.

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The Numerical Solution of the Biharmonic Equation, Using a Spectral Multigrid Method

Cited by 7 publications

References 8 publications

Multigrid Solution of Automatically Generated High-Order Discretizations for the Biharmonic Equation

Multigrid Solution of Automatically Generated High-Order Discretizations for the Biharmonic Equation

Numerical algorithm for the calculation of nonsymmetric dipolar and rotating monopolar vortex structures

Solution of the two dimensional second biharmonic equation with high‐order accuracy

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