Hybrid Mimetic Finite Element Method with Strong Coupling of Continuity Applied to Stokes' Equations
Z. Wu (TU Delft - Aerospace Engineering)
M.I. Gerritsma – Mentor (TU Delft - Aerospace Engineering)
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Abstract
The mimetic spectral element method (MSEM) is a discretization technique that preserves the topological relations and structures presented in PDE's, given certain function spaces and metric-free discrete differential operators that mimic a de Rham or double de Rham complices. This scheme leads to favorable properties in numerical methods, especially conservation of field variables such as mass or momentum.
To numerically solve PDE's, parallelization of the problem has been well studied to improve efficiency. Domain decomposition is one of the methods, which is self explanatory, and it can be applied to MSEM, called hybrid MSEM. In this paper, hMSEM is proposed in the way that variables at the interfaces between or among subdomains are explicitly coupled with the aid of Lagrange multipliers. A downside in this method is that singularity arises when trying to couple variables assigned to edges. The solution to this issue is discussed in this paper that the number of equations of constraints over edges is reduced by 1, while the discrete system still remains rigorous and can be solved in parallel. Such hMSEM is then implemented and applied to Poisson's equation and Stokes' equations for test, to which the accuracy and convergence rate of this method are verified.