Optimal error estimates of an \(H^1\)-Galerkin mixed finite element method for nonlinear Kirchhoff-type problem. (English) Zbl 07830972
Summary: An \(H^1\)-Galerkin mixed finite element method (MFEM) is presented for the nonlinear Kirchhoff-type problem by the bilinear element \(Q_{11}\) and zero order Raviart-Thomas element \(Q_{10}\times Q_{01}\). Optimal error estimates of order \(O(h)\) and \(O(h + \tau^2)\) for the original variable \(u\) in \(H^1\) norm and the flux variable \(\vec{p} = \nabla u\) in \(H(\mathrm{div}; \Omega)\) norm about the semi-discrete scheme and the linearized fully-discrete scheme are derived, respectively. Finally, some numerical results are provided to verify the theoretical analysis.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
Keywords:
nonlinear Kirchhoff-type problem; \(H^1\)-Galerkin MFEM; semi-discrete and fully-discrete schemes; optimal error estimatesReferences:
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