A mixed finite element method for fourth order elliptic equations with variable coefficients.

*(English)*Zbl 0553.65080The Dirichlet problem for a fourth order selfadjoint elliptic operator with variable coefficients on a convex polygonal region is considered. Using the inverse of an appropriate matrix of the coefficients of the operator another variational problem is stated (a mixed method formulation). This new problem is investigated and its relation to the solution of the initial one is shown. Three examples are considered: The biharmonic operator, the bending problem of a clamped orthotropic plate with variable thickness and the isotropic case. The derived problem is approximated by the finite element method using piecewise polynomial basis functions on the triangles of a regular triangulation of the considered region. Error estimates in Sobolev spaces are proposed. In the case of the considered examples this mixed finite element method gives a simultaneous approximation of the displacement and twisting and ”actual” bending moments which is of a great practical importance.

Reviewer: P.S.Vassilevski

##### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

74S05 | Finite element methods applied to problems in solid mechanics |

35J40 | Boundary value problems for higher-order elliptic equations |

74K20 | Plates |

##### Keywords:

mixed finite element method; Dirichlet problem; fourth order; variable coefficients; biharmonic operator; bending problem of a clamped orthotropic plate; Error estimates; bending moments
PDF
BibTeX
XML
Cite

\textit{S. Balasundaram} and \textit{P. K. Bhattacharyya}, Comput. Math. Appl. 10, 245--256 (1984; Zbl 0553.65080)

Full Text:
DOI

##### References:

[1] | Adams, R.A., Sobolev spaces, (1975), Academic Press New York · Zbl 0186.19101 |

[2] | Babuška, I.; Osborn, J.; Pitkäranta, J., Analysis of mixed methods using mesh dependent norms, Math. computation, 35, 152, (1980) · Zbl 0472.65083 |

[3] | Balasundaram, S.; Bhattacharyya, P.K., A mixed finite element method for the Dirichlet problem of fourth order elliptic operators with variable coefficients, () · Zbl 0508.76006 |

[4] | Balasundaram, S.; Bhattacharyya, P.K., Mixed finite element method for fourth order elliptic equations with variable coefficients, Int. conf. on finite element methods, (August (1982)), Shanghai |

[5] | S. Balasundaram and P.K. Bhattacharyya, A mixed finite element method for fourth order partial differential equations. (to appear in ZAMM). · Zbl 0616.73064 |

[6] | S. Balasundaram and P.K. Bhattacharyya, A mixed finite element method for fourth order elliptic problems with variable coefficients. (Paper communicated for publication). · Zbl 0647.65075 |

[7] | Balasundaram, S.; Bhattacharyya, P.K., On existence of solution of the Dirichlet problem of fourth order partial differential equations with variable coefficients, Quart. appl. math., 39, (1983) · Zbl 0533.35024 |

[8] | Brezzi, F.; Raviart, P.A., Mixed finite element methods for 4th order elliptic equations, () · Zbl 0434.65085 |

[9] | Ciarlet, P.G., The finite element method for elliptic problems, (1977), North-Holland Amsterdam |

[10] | Falk, R.S.; Osborn, J.E., Error estimates for mixed methods, R.a.i.r.o., 14, 3, (1980) · Zbl 0467.65062 |

[11] | Hellan, K., Analysis of elastic plates in flexure by a simplified finite element method, Acta polytechnica scandinavica, (1967), Civil Engineering Series No. 46, Trondheim · Zbl 0237.73046 |

[12] | Hermann, L., Finite element bending analysis for plates, J. engng mech. div. ASCE EM5, 93, (1967) |

[13] | Huber, M.T., Probleme der statik technischen wichtiger orthotroper platten, (1929), Warsaw |

[14] | Kondratev, V.A., Boundary value problems for elliptic equations in domains with conical or angular points, Trudy mosk. mat. obsc., 16, (1967) |

[15] | Lekhnitskii, S.G., Anisotropic plates, (1968), Gordon & Breach Science Publishers New York · Zbl 0104.19101 |

[16] | Lions, J.L., Problème aux limites dans LES equations Dérives partielles, (1965), Les Presses de l’Université de Montréal Montréal |

[17] | Miyoshi, T., A finite element method for the solution of fourth order partial differential equations, Kumamoto J. sci. (math.), 9, (1973) |

[18] | Oden, J.T.; Carey, G.F., Finite elements—mathematical aspects, IV, (1983), The Texas Finite Element Series. Prentice Hall Englewood Cliffs, New Jersey |

[19] | Zienkiewicz, O.C., The finite element method, (1977), McGraw-Hill New York · Zbl 0435.73072 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.