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In this paper, we propose a PDE-based optimization motivated by the problemof microfluidic heat transfer to finding the optimal incompressible velocityfields in 2D domain. To solve this optimization model, we use spectral methodto discretize the model to obtain an ODE based optimization. This waysignificantly reduces the complexity of the discretization optimization model,and gives a more accurate approximation of the original PDE based optimization.Some theoretical results are obtained.

PDE-Based Optimization for Advection Diffusion Equation in 2D Domain