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.
Singular Perturbation; Advection-Diffusion; Robin BVP; Finite Element Method; Shishkin Mesh; Error Estimation1. Introduction
We consider the singularly perturbed advection-diffusion Robin boundary values problem
with sufficiently smooth functions, and a small positive parameter. We assume that be decreasing monotonously, moreover
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
where the regular component u(x) satisfy
while the layer component satisfy
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
(2’)
4. The Analysis of Finite Element Approximation
We consider the Galerkin approximation in form of Find such that
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
For the second term of inequality (8), we make use of the coerciveness, continuousness of and the Galerkin orthogonality relation: to obtain that
Thus
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’)
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]).
REFERENCESNOTESReferencesK. W. Chang and F. A. Howes, “Nonlinear Singular Perturbation Phenomena: Theory and Applications,” Spring-Verlag, New York, 1984.S. C. Brenner and L. R. Scott, “The Mathematical Theory of Finite Element Methods,” Springer, Berlin, 1994. http://dx.doi.org/10.1007/978-1-4757-4338-8H. G. Roos, M. Stynes and L. Tobiska, “Numerical Methods for Singularly Perturbed Differential Equations,” Springer, Berlin, 1996. http://dx.doi.org/10.1007/978-3-662-03206-0R. B. Kellogg and A. Tsan, “Analysis of Some Difference Approximations for a Singular Perturbation Problem without Turning Points,” Mathematics of Computation, Vol. 32, No. 144, 1978, pp. 1025-1039. http://dx.doi.org/10.1090/S0025-5718-1978-0483484-9F. Brezzi, L. D. Marini and A. Russo, “On the Choice of Stabilizing Subgrid for Convection-Diffusion Problems,” Computer Methods in Applied Mechanics and Engineering, Vol. 194, No. 2-5, 2005, pp. 127-148. http://dx.doi.org/10.1016/j.cma.2004.02.022T. Linβ and N. Madden, “A Finite Element Analysis of Coupled System of Singularly Perturbed Reaction-Diffusion Equations,” Applied Mathematics and Computation, Vol. 148, No. 3, 2004, pp. 869-880. http://dx.doi.org/10.1016/S0096-3003(02)00955-4M. K. Kadalbajoo, A. S. Yadaw and D. Kumar, “Comparative Study of Singularly Perturbed Two-Point BVPs via: Fitted-Mesh Finite Difference Method, B-Spline Collocation Method and Finite Element Method,” Applied Mathematics and Computation, Vol. 204, No. 2, 2008, pp. 713-725. http://dx.doi.org/10.1016/j.amc.2008.07.014L. P. Franca and E. G. Dutra do Carmo, “The Galerkin Gradient Least Squares Method,” Computer Methods in Applied Mechanics and Engineering, Vol. 74, No. 1, 1989, pp. 41-54. http://dx.doi.org/10.1016/0045-7825(89)90085-6F. Ilinca and J.-F. Hétu, “Galerkin Gradient Least-Squares Formulations for Transient Conduction Heat Transfer,” Computer Methods in Applied Mechanics and Engineering, Vol. 191, No. 27-28, 2002, pp. 3073-3097. http://dx.doi.org/10.1016/S0045-7825(02)00242-6T. Linβ, “Layer-Adapted Meshes for Convection-Diffusion Problems,” Computer Methods in Applied Mechanics and Engineering, Vol. 192, No. 9-10, 2003, pp. 1061-1105. http://dx.doi.org/10.1016/S0045-7825(02)00630-8M. Stynes, “Steady-State Convection-Diffusion Problems,” Acta Numerica, Vol. 14, 2005, pp. 445-508. http://dx.doi.org/10.1017/S0962492904000261M. Stynes and L. Tobiska, “The SDFEM for a Convection-Diffusion Problem with a Boundary Layer: Optimal Error Analysis and Enhancement of Accuracy,” SIAM Journal on Numerical Analysis, Vol. 41, No. 5, 2003, pp. 1620-1642.Z. Zhang, “Finite Element Superconvergence Approximation of One Dimensional Singularly Perturbed Problems,” Numerical Methods for Partial Differential Equations, Vol. 18, No. 3, 2002, pp. 374-395. http://dx.doi.org/10.1002/num.10001Z. Zhang, “Finite Element Super-Convergence on Shishkin Mesh for 2-d Convection-Diffusion Problems,” Mathematical and Computer Modelling, Vol. 72, No. 243, 2003, pp. 1147-1177. http://dx.doi.org/10.1090/S0025-5718-03-01486-8
, and a small positive parameter
. We assume that 
be decreasing monotonously, moreover
at 
(see [
If
for
, 
,then
for 
such that
, then there holds 
which results in a contradiction to
;Thus
.
the differential operator on 
at 
gives
therefore we can conclude that the minimum of 
is non-negative.
satisfy 
for
, and
,
, then 
for all
.
, then we have
.
be the solution to (1) (2). then there exists a constant C, such that for all
, we have the splitting
satisfy
spontaneously since singular perturbation.
such that 
on 
thus 
on
and then extended on (0,1) with
;
on
, we know that 
satisfy
on 
posses the same properties as
, thus we just make discussion on the simplified problem below
(2’)
such that
, the bilinear form
is chosen by
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.
as our work space, 
denotes the piecewise linear interpolant to 
at some special mesh points on I, We’ll utmost estimate the error
.
and the Galerkin orthogonality relation: 
to obtain that
below.
of (1’), (2’) and its piecewise linear interpolant 
satisfy
, we have correspondingly
To obtain the estimation for singular component, we use a Taylor expansion
,
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’)
and allocate uniformly 
and
. In practice one typically has
, we just acquiesce in this case thus
,
estimation, the standard Aubin-Nitche dual verification skill may be involved.