ABSTRACT

A singularly perturbed advection-diffusion two-point Robin boundary value problem whose solution has a single boundary layer is considered. Based on the piecewise linear polynomial approximation, the finite element method is applied to the problem. Estimation of the error between solution and the finite element approximation is given in energy norm on shishkin-type mesh.

1. Introduction

We consider the singularly perturbed advection-diffusion Robin boundary values problem

(1)

(2)

with sufficiently smooth functions, and a small positive parameter. We assume that be decreasing monotonously, moreover

(3)

which guarantees the unique solvability of the problem. It is well known that there exists a boundary layer of width at (see [1], K.W. Chang & F.A. Howes 1984). Standard numerical methods for singularly perturbed problem exhibit spurious error unless the layeradapted-mesh, such as Shishkin mesh, B-mesh(see [2-7]) are employed, for the solutions of singularly perturbed problem usually contain layers. The main objective of the paper is to use the method of singular perturbation to give the estimation of error between solution and the finite element approximation w.r.t. some energy norm on shishkin-type mesh.

Throughout the paper, we shall use C to denote a generic positive constant ,that is independent of ε and mesh, while it can value differently at different places, we occasionally use a subscribed one such as C₁.

2. Properties of Solution for Continuous Problem

In this section, some properties and bounds of the exact solution and its derivatives are deduced preliminarily.

Lemma 1 (Maximum principle) Let If

for, ,then

for

Proof. Assume that there exists such that

If, then there holds which results in a contradiction to;Thus.

Since we have the differential operator on at gives

which result in a contradiction to therefore we can conclude that the minimum of is non-negative.

Lemma 2 (Comparison principle) If satisfy for, and,

, then for all

.

Lemma 3 (Stability result) If, then we have

for all.

The Proofs of Lemma 2 and Lemma 3 are followed essentially from Lemma 1. (See [3] Roos, Stynes and Tobiska, (1996)).

Lemma 4 Let be the solution to (1) (2). then there exists a constant C, such that for all, we have the splitting

(4)

where the regular component u(x) satisfy

(5)

while the layer component satisfy

. (6)

Proof. It is known that (see [4] Kellogg 1978, Chang & Howes 1984)

We assume spontaneously since singular perturbation.

We set such that

and on thus on

and then extended on (0,1) with;

Next let

Then considering that on, we know that satisfy

on

3. Simplification

For simplification of the original problem, we set a transformation

then Equation (1), (2) are transformed to

Continuing, we transform the boundary values homogeneously by

at last, the problem (1), (2) are converted to

where in the posses the same properties as, thus we just make discussion on the simplified problem below

(1’)

(2’)

4. The Analysis of Finite Element Approximation

We consider the Galerkin approximation in form of Find such that

(7)

where, the bilinear form

And a natural norm associated with is chosen by

wherein

is the usual 2-norm.

It is easy to see that is coercive with respect to by the assumption of the monotony of which guarantees the existence of the solution of (7) (see [8-10]). Let N be an even positive integer that denotes the number of mesh intervals.

We consider the space of piecewise linear function denoted by as our work space, denotes the piecewise linear interpolant to at some special mesh points on I, We’ll utmost estimate the error.

Firstly we have

(8)

For the second term of inequality (8), we make use of the coerciveness, continuousness of and the Galerkin orthogonality relation: to obtain that

Thus

. (9)

Combined with (8), we just need to estimate the interpolation error bound below.

Lemma 5 The solution of (1’), (2’) and its piecewise linear interpolant satisfy

Proof. According to the splitting of, we have correspondingly

From Lemma 1 we have

To obtain the estimation for singular component, we use a Taylor expansion

to express the error bound

Continuously, we use the inequality involved a positive monotonically decreasing function g on

Thus we have

Hence

For the proof of the second statement, we have

thus, lemma 5 follows.

Theorem For, defined before, when the Shishkin mesh are applied ,we have the parameter uniform error bound in the energy norm naturally associated with the weak formulation of (1’), (2’)

(10)

Proof. Firstly, we have by triangle inequality and (9)

where in C’s and C₁ are stated before. thus we have

Now we use the classical Shishkin mesh (see [11-13]) by setting the mesh transition parameter defined by

and allocate uniformly

points in each of and. In practice one typically has, we just acquiesce in this case thus

thus for,

Also for

Combining the above two cases reads (10).

Remark. To obtain estimation, the standard Aubin-Nitche dual verification skill may be involved.

The superconvergence phenomena on Shishkin mesh for the convection-diffusion problems can be discussed according to Z. Zhang (see [13,14]).

REFERENCES

NOTES

^*This research was supported by the National Natural Science Foundation of China (Grant No. 50975003).