# Interdisciplinary Applied Mathematics

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The spatial discretization of p,Flow is based on the spectral element method, which is similar to the hp version of finite element methods (Kar-niadakis and Sherwin, 1999). Typical meshes for simulation in a rough microchannel with different types of roughness (presented in Section 4.1.4) are shown in Figure 14.1. The two-dimensional domain is broken up into elements similar to finite elements, but each element employs high-order interpolants based on Legendre polynomials. The N points that determine the interpolant correspond to locations of maxima of the Legendre polynomials and include the end-points. For smooth solutions, the spatial discretization error decays exponentially to zero (spectral or p-convergence). This means that if we double N, the error will decay by two orders of magnitude. This is a much faster decay than in standard second-order methods, which yield an error reduction by a factor of only four. In microflow simulations, the spectral element method can be used efficiently by exploiting the dual    path    of    convergence    allowed    by    the method.    For    example,    in    re

gions of geometric complexity a finite element-like discretization (low N and small-size elements) can be employed, whereas in regions of homogeneous geometry with steep gradients a spectral-like discretization (high N and large-size elements) can be employed. In particular, the boundary conditions for microflows, either for gases (Knudsen effects) or liquids (electrokinetic effects) can be resolved very accurately. On the other hand, the computational cost of the spectral element method is O(KNd+v), where d = 2 and d =3 for two-dimensional and three-dimensional discretizations, respectively, and K is the number of elements. This is higher compared

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In-Phase Channel with aspect ratio of 20 :1

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