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

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.


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
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