Linear interpolation-based scheme

The momentum and continuity equations give rise to a 22 x 22 elemental stiffness matrix as is shown by Equation (3.31). In Equation (3.31) the subscripts I and / represent the nodes in the bi-quadratic element for velocity and K and L the four corner nodes of the corresponding bi-linear interpolation for the pressure. The weight functions. Nr and Mf, are bi-qiiadratic and bi-linear, respectively. The y th component of velocity at node J is shown as iPj. Summation convention on repeated indices is assumed. The discretization of the continuity and momentum equations is hence based on the U--V- P scheme in conjunction with a Taylor-Hood element to satisfy the BB condition. 

TABLE 2—Relative errors for DS, FFT-based method and MLMI over different grids (%) (Green, Constant and Bilinear stand, respectively, for the schemes based on Green s function, constant function, and linear interpolation in determining the influence coefficients). ... 

The interfacial value of thermal conductivity can be best interpolated by invoking a harmonic mean approximation, based on values of the same at the adjacent grid points. The explicit scheme gives rise to a system of discretized algebraic equations that are not mathematically coupled. However, the cost that one might have to pay against this simplification is that the scheme is conditionally stable. On the other hand, the implicit scheme requires a coupled system of linear algebraic equations to be solved but is unconditionally stable (the issues of stability in the context of discretized equations will be elaborated later). 

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