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Iterative method for solving the Neumann boundary value problem for biharmonic type equation. (English) Zbl 1101.65101

The solution of boundary value problems (BVPs) for fourth order differential equations by their reduction to BVP for second order equations, with the aim to use the achievements for the latter ones attracts attention from many researchers. Using the technique developed by the author in recent works, he constructs an iterative method for the Neumann BVP for a biharmonic type equation. The convergence rate of the method is proved and some numerical experiments are performed for testing it in dependence on the choice of an iterative parameter.

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
35J40 Boundary value problems for higher-order elliptic equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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References:

[1] Abramov, A. A.; Ulijanova, V. I., On a method for solving biharmonic type equation with singularly small parameter, J Comput. Math. Math. Phys., 32, 4, 567-575 (1992), (Russian)
[2] Aubin, J. P., Approximation of Elliptic Boundary-value Problems (1971), Wiley-Interscience: Wiley-Interscience New York
[3] Begis, D.; Perronet, A., The club modulef, (Marchuc, G.; Lions, J. L., Numerical Methods in Applied Mathematics (1982), Nauka: Nauka Novosibirsk), 212-236, (Russian) · Zbl 0566.65077
[4] Dang, Q. A., On an iterative method for solving a boundary value problem for fourth order differential equation, Math. Phys. Nonlinear Mech., 10, 44, 54-59 (1988), (Russian)
[5] Dang, Q. A., Approximate method for solving an elliptic problem with discontinuous coefficients, J. Comput. Appl. Math., 51, 2, 193-203 (1994) · Zbl 0808.65130
[6] Dang, Q. A., Boundary operator method for approximate solution of biharmonic type equation, J. Math., 22, 1& 2, 114-120 (1994) · Zbl 0940.65522
[7] Dang, Q. A., Mixed boundary-domain operator in approximate solution of biharmonic type equation, Vietnam J. Math., 26, 3, 243-252 (1998) · Zbl 0939.35061
[8] Dorodnisyn, A.; Meller, N., On some approaches to the solution of the stationary Navier-Stoke equation, J. Comput. Math. Math. Phys., 8, 2, 393-402 (1968), (Russian) · Zbl 0197.24901
[9] Glowinski, R.; Lions, J-L.; Tremoliere, R., Analyse numerique des inequations variationelles (1976), Dunod: Dunod Paris
[10] Lions, J.-L.; Magenes, E., Problemes aux limites non homogenes et applications, vol. 1 (1968), Dunod: Dunod Paris · Zbl 0165.10801
[11] Palsev, B. V., On the expansion of the Dirichlet problem and a mixed problem for biharmonic equation into a seriaes of decomposed problems, J. Comput. Math. Math. Phys., 6, 1, 43-51 (1966), (Russian)
[12] Samarskii, A. A., The Theory of Difference Schemes (2001), Marcel Dekker: Marcel Dekker New York · Zbl 0971.65076
[13] Samarskii, A.; Nikolaev, E., Numerical Methods for Grid Equations, vol. 2 (1989), Birkhäuser: Birkhäuser Basel
[14] Young, D. M., Iterative Solution of Large Linear Systems (1971), Academic Press: Academic Press New York · Zbl 0204.48102
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