The Dual Mesh Finite Domain Method for the Convection–Diffusion Heat Equation and the Navier–Stokes

Matthew A. Martinez and J. N. Reddy


The Dual Mesh Finite Domain Method (DMFDM), which was introduced in [1], divides a domain, Ω into grid of non-overlapping subdomains called the primal finite element mesh, or sometimes the primal mesh for short. This primal mesh of finite elements is then overlaid by a secondary mesh of node-centered control domains, which may occupy more than one finite element. The integral form of the equation, as opposed to a weak form or wegihted-residual statement, is evaluated over each node-centered control domain. In evaluating these integral forms, the values of primary and/or secondary variables are needed at the extremes of the control domain. The DMFDM takes advantage of the primary mesh of finite elements and uses their interpolation functions to provide values at the extremes of each control domain.



The discretiztion for a 1-D domain of linear and quadratic primal elements are shown in Figures 1 and 2, respectively. The 1-D results can be seen presented in [2].


Fig. 1: Linear interpolations on both sides of the Ith control domain.


The same method is then expanded to 2-D, which can be seen in Figures 3 and 4. It is especially important to note that, for the quadratic grid, providing the coefficients for the Ith node will dpepend on the node’s location within the primal mesh. The value of the Ith node may depend on anywhere from 8 to 24 other nodes.


Fig. 2: Stencil of the Ith control domain about a node on the border of two elements in the quadratic primal mesh.



Fig. 3: Stencil for the linear 2-D DMFDM elements surrounding interior node I.



Fig. 4: Stencil for the quadratic 2-D DMFDM elements surrounding interior node I.




Fig. 5: Comparison between linear and quadratic 2D DMFDM and the Upwind and Power Law Finite Volume Methods.



Fig. 6: Results for the 2-D linear DMFDM for the Navier–Stokes solution of the Lid Driven Cavity Problem.



Fig. 7: Results for the 2-D linear DMFDM for the Navier–Stokes solution fo the flow over the backwards step.



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