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Chebyshev spectral methods for potential field computation Zhao, Shengkai


We investigate Chebyshev spectral methods for solving Poisson's equation and the generalized Poisson's equations and apply them to 3-D gravity and DC resistivity modeling. When we approximate a function by a finite sum of Chebyshev polynomials, there are two kinds of errors: truncation and aliasing. We present a cell-average discretization scheme to reduce the aliasing error. Both theoretical analysis and numerical examples show that when there are discontinuities in a function, the cell-average discretization is better than the point injection discretization. We use both the r and collocation methods to solve Poisson's equation V2u = f. We solve the discrete systems by a matrix-diagonalization method. The speed and accuracy of collocation methods are better than those of the r method. For the generalized Poisson's equation V • (oVu) = f, we present a new iterative method. We rewrite the equation in the form of a Poisson's equation, V2u = (f —Va•Vu)la. At each iteration we compute the right hand-side term from the current value of u first. Then we solve the resultant Poisson’s equation by the collocation method. Numerical results show that the convergence rate of the new method is much faster than that of the spectral multi grid method. When there are discontinuities in the source function f and/or the conductivity o, the single-domain Chebyshev spectral method does not converge exponentially. In this case we use the multi-domain Chebyshev spectral method to solve the problem. We divide the whole domain into a number of subdomains so that in each subdomain the function is infinitely differentiable. Then we approximate the function by a separate Chebyshev series in each subdomain. We determine the relations between the Chebyshev polynomials in adjacent subdomains by the interface conditions. In this way we can achieve exponential convergence. In 3-D gravity modeling, we present a method to realize 2-D and 3-D point-injection and cell-average discretizations, use a multi pole expansion to compute the approximate boundary conditions and solve the equations by both single-domain and multi-domain Chebyshev spectral methods. We also extend Okabe's analytic formulation for the gravitational field of a homogeneous, polyhedral body to the potential. Numerical results show that the accuracy of the cell-average discretization is better than that of the point injection discretization. The multi-domain solution is the best. We apply the multi-domain Chebyshev spectral method to 3-D DC resistivity modeling. We discuss the singularity removal and present a modified finite-difference method for comparison. For a two-layered model, the multi-domain Chebyshev spectral solution is far more accurate than the finite-difference solution. For piecewise-constant and piecewise-smooth models the solutions obtained by both methods agree with each other quite well. However, the Chebyshev spectral method is more efficient than the finite-difference method.

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