Abstract
It is well known that the Ciarlet-Raviart mixed method for solving the biharmonic eigenvalue problem with Navier Boundary Condition by decomposing the operator into two Laplacians may generate spurious eigenvalues on non-convex domains. To overcome this difficulty, we adopt a recently developed mixed method, which decomposes the biharmonic equation into three Poisson equations and still recovers the original solution. Using this idea, we design an efficient biharmonic eigenvalue algorithm, which contains only Poisson solvers. With this approach, eigenfunctions can be confined in the correct space and thereby spurious modes in non-convex domains are avoided. A priori error estimates for both eigenvalues and eigenfunctions on quasi-uniform meshes are obtained; in particular, a convergence rate of \({\mathcal {O}}({h}^{2\alpha })\) (\( 0<\alpha <\pi /\omega \), \(\omega > \pi \) is the angle of the reentrant corner) is proved for the linear finite element. Surprisingly, numerical evidence demonstrates an \({\mathcal {O}}({h}^{2})\) convergent rate for the quasi-uniform mesh with the regular refinement strategy even on non-convex polygonal domains.
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Acknowledgements
The authors would like to thanks anonymous referees for their valuable comments. This work is supported in part by the National Natural Science Foundation of China grants NSFC 11871092, 12131005, and NSAF U1930402. H. Li was partially supported by the NSF Grant DMS-1819041.
Funding
This work is supported in part by the National Natural Science Foundation of China grants NSFC 11871092, 12131005, and NSAF U1930402. H. Li was partially supported by the NSF Grant DMS-1819041.
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Zhang, B., Li, H. & Zhang, Z. Solving Biharmonic Eigenvalue Problem With Navier Boundary Condition Via Poisson Solvers On Non-Convex Domains. J Sci Comput 92, 24 (2022). https://doi.org/10.1007/s10915-022-01878-9
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DOI: https://doi.org/10.1007/s10915-022-01878-9