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A Weak Formulation of the Immersed Boundary Method

A new method of spatial discretization for immersed boundary computations is introduced. Fluid velocity and pressure are obtained as weak solutions of the discretized fluid equations with respect to a wavelet basis of functions. The scaling function of the fluid velocity basis may be chosen to be identical to Peskin's discrete approximation to the Dirac delta function. On a regular rectangular grid the discretized equations are solved using the fast Fourier transform, retaining the efficiency of the immersed boundary method. We show experimental numerical evidence that the rate of volume loss of our method is better than that of the finite difference immersed boundary method. Our formulation offers new insights into the immersed boundary method and leads to new extensions and applications.