An optimal \(C^0\) finite element algorithm for the 2D biharmonic problem: Theoretical analysis and numerical results. (English) Zbl 0997.65133

Amara, M.; Dabaghi, F.

Numer. Math. 90, No. 1, 19-46 (2001).

This paper analyzes a new mixed finite element method for the numerical approximation of the biharmonic problem. This method is based on the Ciarlet-Raviart ones and has the same numerical properties as the Glowinski-Pironneau ones.
The main improvement consists to replace the usual bilinear form \(a(.,.)\) by an approximated form \(a_h(.,.)\) with nice ellipticity properties. Convergence and a priori error estimates are mathematically studied and optimal with respect to the regularity of the solution.
Some numerical simulations complete the paper and illustrate the efficiency and the optimality of the method. This method can be extended to bi-dimensional Navier-Stokes equations, plate equations and plane crack propagation problems. Moreover, the presentation of the paper is clear and attractive.

Reviewer: Michel Bernadou (Paris La Defense)

MSC:

65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
31A30	Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions
35J40	Boundary value problems for higher-order elliptic equations
74S05	Finite element methods applied to problems in solid mechanics
65N12	Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15	Error bounds for boundary value problems involving PDEs
35Q30	Navier-Stokes equations
74K20	Plates
74R10	Brittle fracture

Keywords:

biharmonic problem; mixed finite element method; optimal a priori error estimates; numerical results; Ciarlet-Raviart method; Glowinski-Pironneau method; convergence; Navier-Stokes equations; plate equations; crack propagation