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The fictitious domain method with L 2 ‐penalty for the Stokes problem with the Dirichlet boundary condition
Author(s) -
Zhou Guanyu
Publication year - 2018
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22235
Subject(s) - mathematics , norm (philosophy) , penalty method , discretization , rate of convergence , mathematical analysis , dirichlet boundary condition , finite element method , dirichlet problem , stokes problem , boundary value problem , stokes flow , mathematical optimization , geometry , physics , law , channel (broadcasting) , flow (mathematics) , political science , thermodynamics , electrical engineering , engineering
We consider the fictitious domain method with L 2 ‐penalty for the Stokes problem with the Dirichlet boundary condition. First, we investigate the error estimates for the penalty method at the continuous level. We obtain the convergence of order O ( ϵ 1 4) in H 1 ‐norm for the velocity and in L 2 ‐norm for the pressure, where ϵ is the penalty parameter. The L 2 ‐norm error estimate for the velocity is upgraded to O ( ϵ ) . Moreover, we derive the a priori estimates depending on ϵ for the solution of the penalty problem. Next, we apply the finite element approximation to the penalty problem using the P1/P1 element with stabilization. For the discrete penalty problem, we prove the error estimate O ( h + ϵ 1 4) in H 1 ‐norm for the velocity and in L 2 ‐norm for the pressure, where h denotes the discretization parameter. For the velocity in L 2 ‐norm, the convergence rate is improved to O ( h + ϵ 1 2) . The theoretical results are verified by the numerical experiments.