Wavelet-based methods have demonstrated great potential for solving partial differential equations of various types. The capabilities of the wavelet Galerkin method are explored by solving various heat transfer and fluid flow problems. A fictitious domain approach is used to simplify the discretization of the domain and a penalty method allows an efficient implementation of the boundary conditions. The resulting system of equation is solved iteratively via the Conjugate Gradient and Preconditioned Conjugate Gradient Methods. The fluid flow problems in the present study are formulated in such a manner that the solution of the continuity and momentum equations is obtained by solving a series of Poisson equations. This is achieved by using steepest descent method. The examples solved show that the method is amenable to solving large problems rapidly with modest computational resources.

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