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A spectral method for elliptic equations: the Dirichlet problem Kendall Atkinson · David Chien · Olaf Hansen

Received: 3 September 2008 / Accepted: 8 March 2009 / Published online: 16 April 2009 © Springer Science + Business Media, LLC 2009

Abstract Let  be an open, simply connected, and bounded region in Rd , d ≥ 2, and assume its boundary ∂ is smooth. Consider solving an elliptic partial differential equation Lu = f over  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 un of degree ≤ n that is convergent to u. The transformation from  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 ∈ C∞ () and assuming ∂ is a C∞ boundary, the convergence of u − un  H1 to zero is faster than any power of 1/n. Numerical examples in R2 and R3 show experimentally an exponential rate of convergence. Keywords Spectral method · Elliptic equations · Dirichlet problem · Galerkin method Mathematics Subject Classification (2000) 65N35

Communicated by Yuesheng Xu. K. Atkinson (B) Departments of Mathematics & Computer Science, The University of Iowa, Iowa City, Iowa, USA e-mail: [email protected] D. Chien · O. Hansen Department of Mathematics, California State University San Marcos, San Marcos, CA, USA

170

K. Atkinson et al.

1 Introduction Consider solving the elliptic partial differential equation   d  ∂ ∂u(s) ai, j(s) + γ (s)u(s) = f (s), Lu(s) ≡ − ∂si ∂sj i, j=1

s ∈  ⊆ Rd

(1)

with the Dirichlet boundary condition u(s) ≡ 0,

s ∈ ∂

(2)

Assume d ≥ 2. Let  be an open, simply-connected, and bounded region in Rd , and assume that its boundary ∂ is smooth and sufficiently differentiable. Similarly, assume the functions γ (s), f (s), ai, j(s) are several timescontinuously differentiable over . As usual, assume the matrix A(s) = ai, j(s) is symmetric and satisfies the strong ellipticity condition, ξ T A(s)ξ ≥ c0 ξ T ξ,

s ∈ ,

ξ ∈ Rd

(3)

with c0 > 0. Also assume γ (s) ≥ 0, s ∈ . In Section 2 we consider the special region  = B, the open unit ball in Rd . We define a Galerkin method for (1)–(2) with a special finite-dimensional subspace of polynomials, and we give an error analysis that shows rapid convergence of the method. In Section 3 we discuss the use of a transformation from a general region  to the unit ball B, showing that the transformed equation is again elliptic over B. Implementation issues are discussed in Section 4 for problems in R2 and R3 . We conclude in Section 5 with numerical examples in R2 and R3 . The methods of this paper generalize to the equation   d  ∂u(s) ∂ ai, j(s) Lu(s) ≡ − ∂si ∂sj i, j=1 +

d  j=1

bj (s)

∂u(s) + γ (s)u(s) = f (s), ∂sj

s ∈  ⊆ Rd

which contains first order derivative terms, provided the operator L is strongly elliptic. To do so, use the results given in Brenner and Sco