Abstract
The Nitsche method is a method of “weak imposition” of the inhomogeneous Dirichlet boundary conditions for partial differential equations. This paper explains stability and convergence study of the Nitsche method applied to evolutionary diffusion–advection-reaction equations. We mainly discuss a general space semidiscrete scheme including not only the standard finite element method but also Isogeometric Analysis. Our method of analysis is a variational one that is a popular method for studying elliptic problems. The variational method enables us to obtain the best approximation property directly. Actually, results show that the scheme satisfies the inf-sup condition and Galerkin orthogonality. Consequently, the optimal order error estimates in some appropriate norms are proven under some regularity assumptions on the exact solution. We also consider a fully discretized scheme using the backward Euler method. Numerical example demonstrate the validity of those theoretical results.
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Acknowledgements
We thank the anonymous reviewer for his/her valuable comments and suggestions to improve the quality of the paper. This study was supported by JST CREST Grant Number JPMJCR15D1 and JSPS KAKENHI Grant Number 15H03635. The first author was also supported by the Program for Leading Graduate Schools, MEXT, Japan.
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Ueda, Y., Saito, N. The inf-sup condition and error estimates of the Nitsche method for evolutionary diffusion–advection-reaction equations. Japan J. Indust. Appl. Math. 36, 209–238 (2019). https://doi.org/10.1007/s13160-018-0338-4
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DOI: https://doi.org/10.1007/s13160-018-0338-4