Finite Element Analysis for Singularly Perturbed Advection-Diffusion Robin Boundary Values Problem ()
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 ![](https://www.scirp.org/html/8-5300376\e16a889e-0a0f-46ab-85b0-f1f20ecb62bd.jpg)
Proof. Assume that there exists
such that
![](https://www.scirp.org/html/8-5300376\886b9281-f1cc-4856-9ff2-a1753fe209ba.jpg)
If
, then there holds
which results in a contradiction to
;Thus
.
Since we have
the differential operator on
at
gives
![](https://www.scirp.org/html/8-5300376\0e188717-a578-4d63-b9cf-ece42cea4de1.jpg)
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
![](https://www.scirp.org/html/8-5300376\c255a213-3703-4307-9186-1b0414404123.jpg)
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)
![](https://www.scirp.org/html/8-5300376\84f2a286-0587-4fc3-8f51-d181c02761fd.jpg)
We assume
spontaneously since singular perturbation.
We set
such that ![](https://www.scirp.org/html/8-5300376\754ba16c-d2e9-4b53-85fd-442e353e309e.jpg)
and
on
thus
on
and then extended on (0,1) with
;
Next let
![](https://www.scirp.org/html/8-5300376\1aca9c26-a689-4c89-bd0c-1c55bcf60b7f.jpg)
Then considering that
on
, we know that
satisfy
on ![](https://www.scirp.org/html/8-5300376\8cb33733-64c1-46af-a26e-2f38d2866889.jpg)
3. Simplification
For simplification of the original problem, we set a transformation
![](https://www.scirp.org/html/8-5300376\ae35dfbb-cf45-4ac1-b671-720181e0cd35.jpg)
then Equation (1), (2) are transformed to
![](https://www.scirp.org/html/8-5300376\6fed2671-fa97-43b0-a195-cce246bfd63a.jpg)
![](https://www.scirp.org/html/8-5300376\3988e8ec-04ef-4d96-9cd6-63c548523d51.jpg)
![](https://www.scirp.org/html/8-5300376\0fd0ac98-aaf4-4bb2-8732-374d8f96ec11.jpg)
Continuing, we transform the boundary values homogeneously by
![](https://www.scirp.org/html/8-5300376\502bd5c6-ea4d-44a8-b7ac-5e8a42a2091d.jpg)
at last, the problem (1), (2) are converted to
![](https://www.scirp.org/html/8-5300376\e4575355-ad10-4097-be53-83d459ffaf13.jpg)
![](https://www.scirp.org/html/8-5300376\dc825522-4704-4da3-8161-8a890beb134a.jpg)
![](https://www.scirp.org/html/8-5300376\9cfe479f-54c8-4d9d-b0e3-0c7d5e05e291.jpg)
where in the
posses the same properties as
, thus we just make discussion on the simplified problem below
(1’)
![](https://www.scirp.org/html/8-5300376\8bf06e74-02d3-4a84-b76a-0319ff9e9748.jpg)
(2’)
4. The Analysis of Finite Element Approximation
We consider the Galerkin approximation in form of Find
such that
(7)
where
, the bilinear form
![](https://www.scirp.org/html/8-5300376\b4d0a8db-8b42-4282-b701-d4acbe03bdfc.jpg)
And a natural norm associated with
is chosen by
![](https://www.scirp.org/html/8-5300376\db41c38c-3022-49ed-950a-55eb0b41a04b.jpg)
wherein
![](https://www.scirp.org/html/8-5300376\1b48d757-1844-4bb4-bfbb-ae0fdb07f56e.jpg)
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
![](https://www.scirp.org/html/8-5300376\1a31b111-8da4-46e5-8023-98e98a5be035.jpg)
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
![](https://www.scirp.org/html/8-5300376\9ce53d1b-fac3-460b-832c-7bea39a01ccc.jpg)
![](https://www.scirp.org/html/8-5300376\f2a4c5fe-e57c-450e-bf19-e76c1393bad1.jpg)
Proof. According to the splitting of
, we have correspondingly
![](https://www.scirp.org/html/8-5300376\95e3327e-5470-4803-8205-1e6fb8db2023.jpg)
From Lemma 1 we have
To obtain the estimation for singular component, we use a Taylor expansion
![](https://www.scirp.org/html/8-5300376\ba2ec5b2-9d73-4f40-9718-e70998438da9.jpg)
to express the error bound
![](https://www.scirp.org/html/8-5300376\8e45ac4b-d52f-40ae-abcf-0f9caa7a5fc0.jpg)
Continuously, we use the inequality involved a positive monotonically decreasing function g on ![](https://www.scirp.org/html/8-5300376\da2bfd4a-6b56-41fd-ad65-fd03332d0f64.jpg)
![](https://www.scirp.org/html/8-5300376\c9baf648-74c8-4c92-b1b1-06026ea8ffa2.jpg)
Thus we have
![](https://www.scirp.org/html/8-5300376\b34fa1f5-2588-4480-86bb-6397ca18dd31.jpg)
Hence
![](https://www.scirp.org/html/8-5300376\2765f261-a244-41a3-981c-8ca9a3df235d.jpg)
For the proof of the second statement, we have
![](https://www.scirp.org/html/8-5300376\264df40a-f9c1-473b-b45a-2233baf360f2.jpg)
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)
![](https://www.scirp.org/html/8-5300376\e4edeb8f-417b-4fff-8918-7a008d935aa5.jpg)
![](https://www.scirp.org/html/8-5300376\1e8ddf2e-235f-4379-8dd1-6b2b59b0e54d.jpg)
where in C’s and C1 are stated before. thus we have
![](https://www.scirp.org/html/8-5300376\3dfb5327-a0d4-42ae-a9d9-6f32602865f9.jpg)
Now we use the classical Shishkin mesh (see [11-13]) by setting the mesh transition parameter defined by
and allocate uniformly ![](https://www.scirp.org/html/8-5300376\780be3c1-370a-4087-b26a-f74c36cc3061.jpg)
points in each of
and
. In practice one typically has
, we just acquiesce in this case thus
![](https://www.scirp.org/html/8-5300376\8e79aa8e-aed3-48f5-854a-036b72847a1d.jpg)
![](https://www.scirp.org/html/8-5300376\d2e16fa7-f067-4955-bdbe-2aec9aa229f7.jpg)
![](https://www.scirp.org/html/8-5300376\28b3eb79-cda3-4a2d-95db-75a6ff057cdc.jpg)
![](https://www.scirp.org/html/8-5300376\cea6d1dd-709f-4b21-a209-d5372e3e595c.jpg)
thus for
,
![](https://www.scirp.org/html/8-5300376\cdc53103-3882-4840-bfca-dadfd7b92fba.jpg)
Also for ![](https://www.scirp.org/html/8-5300376\4a6ec8ed-0561-443a-a150-e044f7332db4.jpg)
![](https://www.scirp.org/html/8-5300376\67d88cf2-749e-4342-8661-45ed4ad0f843.jpg)
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]).
NOTES