A spectral method for elliptic equations: the Dirichlet problem

作者：Atkinson Kendall; Chien David; Hansen Olaf

来源：Advances in Computational Mathematics, 2010, 33(2): 169-189.

DOI：10.1007/s10444-009-9125-8

• Spectral method
• Elliptic equations
• Dirichlet problem
• Galerkin method

Let Omega be an open, simply connected, and bounded region in R(d), d >= 2, and assume its boundary partial derivative Omega is smooth. Consider solving an elliptic partial differential equation Lu = f over Omega with zero Dirichlet boundary values. The problem is converted to an equivalent elliptic problem over the unit ball B; and then a spectral Galerkin method is used to create a convergent sequence of multivariate polynomials u(n) of degree <= n that is convergent to u. The transformation from Omega to B requires a special analytical calculation for its implementation. With sufficiently smooth problem parameters, the method is shown to be rapidly convergent. For u is an element of C(infinity)((Omega) over bar) and assuming partial derivative Omega is a C(infinity) boundary, the convergence of parallel to u - u(n)parallel to(H1) to zero is faster than any power of 1/n. Numerical examples in R(2) and R(3) show experimentally an exponential rate of convergence.

• 出版日期2010-8