In this work, we use a symbolic algebra package to derive a family of finite difference approximations for the biharmonic equation on a 9-point compact stencil. The solution and its first derivatives are carried as unknowns at the grid points. Dirichlet boundary conditions are thus incorporated naturally. Since the approximations use the 9-point compact stencil, no special formulas are needed near the boundaries. Both second-order and fourth-order discretizations are derived.The fourth-order approximations produce more accurate results than the 13-point classical stencil or the commonly used system of two second-order equations coupled with the boundary condition.The method suffers from slow convergence when classical iteration methods such as Gauss-Seidel or SOR are employed. In order to alleviate this problem we propose several multigrid techniques that exhibit grid-independent convergence and solve the biharmonic equation in a small amount of computer time. Test results from three different problems, including Stokes flow in a driven cavity, are reported.
A line integral is defined as the integral of two-dimensional data along a (onedimensional, straight) line of given length and orientation. Line integrals are used in various forms of edge and line detectors in images and in the computation of the Radon transform. We present a recursive algorithm which enables approximation of discretized line integrals at all lengths, orientations, and locations to within a prescribed error bound in at most O(n log n log log n) operations, where n is the number of data points. Furthermore, for most applications (in particular, where even small amounts of noise are present in the data) all of these integrals can be computed to the desired accuracy in about 24n log n operations.
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