A parameter-uniform numerical method for a singularly perturbed two parameter elliptic problem
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A parameter-uniform numerical method for a singularly perturbed two parameter elliptic problem E. O’Riordan · M. L. Pickett
Received: 15 September 2009 / Accepted: 10 May 2010 / Published online: 1 June 2010 © Springer Science+Business Media, LLC 2010
Abstract In this paper, a class of singularly perturbed elliptic partial differential equations posed on a rectangular domain is studied. The differential equation contains two singular perturbation parameters. The solutions of these singularly perturbed problems are decomposed into a sum of regular, boundary layer and corner layer components. Parameter-explicit bounds on the derivatives of each of these components are derived. A numerical algorithm based on an upwind finite difference operator and a tensor product of piecewise-uniform Shishkin meshes is analysed. Parameter-uniform asymptotic error bounds for the numerical approximations are established. Keywords Singularly perturbed · Two parameter · Elliptic · Shishkin mesh Mathematics Subject Classification (2010) 65N15
Communicated by Martin Stynes. E. O’Riordan (B) School of Mathematical Sciences, Dublin City University, Dublin 9, Ireland e-mail: [email protected] M. L. Pickett Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland
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E. O’Riordan, M.L. Pickett
1 Introduction Consider the following class of singularly perturbed elliptic problems posed on the unit square := (0, 1)2 : find u(x, y) such that Lε,μ u := εu + μa · ∇u − b u = f (x, y), (x, y) ∈ , u = 0,
(x, y) ∈ ∂,
a := (a1 (x), a2 (y)), ¯ a ≥ (α, α) > (0, 0) , b (x, y) ≥ β > 0, (x, y) ∈ , ∂k f ∂k f (1, 0) = i j (0, 1) = 0, 0 ≤ k ≤ 4, i j ∂x ∂y ∂x ∂y f (0, 0) = 0 , αμ2 ≥ γ ε, γ := min ¯
b b , 2a1 2a2
∂k f (1, 1) = 0, 0 ≤ k ≤ 7, ∂ xi ∂ y j
, 0 < ε ≤ 1, 0 < μ ≤ μ0 ,
(1a) (1b) (1c) (1d) (1e) (1f) (1g)
where a1 , a2 , b , f are smooth functions and μ0 is a sufficiently small constant. For sufficiently compatible and smooth boundary data, there is no loss in ¯ generality in assuming zero boundary data. For u to be in C3,ς (), 0< ¯ and ς ≤ 1 it suffices [5, Theorem 3.2] that a1 , a2 , b are smooth, f ∈ C1,ς () f (0, 0) = f (0, 1) = f (1, 0) = f (1, 1) = 0. Since a1 ≥ α > 0, a2 ≥ α > 0, there are no characteristic layers [6, 7] present in the solution. In general, the solution will contain exponential layers [18] with different widths in the vicinity of the sides and the corners of the domain. Classical numerical methods are inappropriate for singularly perturbed problems [4, 18]. A numerical method is said to be parameter-uniform [4, 18], if an error bound of the form U − u N ≤ CN − p , p > 0, exists, where U is the numerical approximation generated on a mesh N , N is the number of mesh elements used in each coordinate direction, v D := max(x,y)∈D |v(x, y)| is the maximum pointwise norm and, crucially, the error constant C is independent of all perturbation paremeters present in the differential equation. To establish parameter-uniform convergence, we derive pointwise bounds on the derivat
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