[00547] Fictitious domain methods with finite elements and penalty over spread interface
Session Time & Room : 5C (Aug.25, 13:20-15:00) @E705
Type : Contributed Talk
Abstract : We present a spread interface approach in fictitious domain methods to decipher the elliptic PDEs depicted over curved complex domains. In this approach, we employ the $L^2$ penalty for a small tubular neighborhood $\Omega_{\delta}$ near $\partial\Omega$ in $\mathrm{R}\backslash\Omega$ in place of the substantial penalty for the whole fictitious part $\mathrm{R}\backslash\Omega$. We achieve strong convergence results concerning the penalty parameter $\epsilon$ in addition to the a priori estimates and stability analysis. We implement the linear finite elements and acquire the expected error estimates. The comprehensive numerical investigations support the mathematical findings, which also anticipate optimal convergence regardless of the convexity and shape of the domain.
Keywords: Fictitious domain methods, Elliptic problems, Curved domain, Error estimates, Uniform mesh.
References:
1. S. Kale, and D. Pradhan, An augmented interface approach in fictitious domain methods, Comput. Math. with Appl., Vol. 125, pp. 238-247, (2022).
2. B. Maury, Numerical Analysis of a finite element/volume penalty method, SIAM J. Numer Anal., Vol. 47(2), (2009), pp. 1126-1148.
3. N. Saito and G. Zhou, Analysis of the fictitious domain method with an $L^2$-penalty for elliptic problems, Numer. Funct. Anal. Optim., Vol. 36, (2015), pp. 501-527.
4. S. Zhang, Analysis of finite element domain embedding methods for curved domains using uniform grids, SIAM J. Numer. Anal., Vol. 46(6), (2008), pp. 2843-2866.
