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A UNIFORMLY CONVERGENT NUMERICAL METHOD FOR A WEAKLY COUPLED SYSTEM OF SINGULARLY PERTURBED CONVECTION-DIFFUSION PROBLEMS WITH BOUNDARY AND WEAK INTERIOR LAYERS†
A UNIFORMLY CONVERGENT NUMERICAL METHOD FOR A WEAKLY COUPLED SYSTEM OF SINGULARLY PERTURBED CONVECTION-DIFFUSION PROBLEMS WITH BOUNDARY AND WEAK INTERIOR LAYERS†
Journal of Applied Mathematics & Informatics. 2015. Sep, 33(5_6): 635-648
Copyright © 2015, Korean Society of Computational and Applied Mathematics
  • Received : February 05, 2015
  • Accepted : June 01, 2015
  • Published : September 30, 2015
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About the Authors
SHEETAL CHAWLA
S. CHANDRA SEKHARA RAO

Abstract
We consider a weakly coupled system of singularly perturbed convection-diffusion equations with discontinuous source term. The diffusion term of each equation is associated with a small positive parameter of different magnitude. Presence of discontinuity and different parameters creates boundary and weak interior layers that overlap and interact. A numerical method is constructed for this problem which involves an appropriate piecewise uniform Shishkin mesh. The numerical approximations are proved to converge to the continuous solutions uniformly with respect to the singular perturbation parameters. Numerical results are presented which illustrates the theoretical results. AMS Mathematics Subject Classification : 65L10, 65L11, 65L12.
Keywords
1. Introduction
An extensive research had been done on numerical methods for a single singularly perturbed convection-diffusion differential equation [1] - [4] , but for system of equations very few works had been done. The classical numerical methods fail to produce good approximations for singularly perturbed problems. Various non-classical approaches produce better approximations and converge uniformly with respect to the small perturbation parameter. In the literature [7] - [14] methods were available to obtain numerical approximation for system of singularly perturbed convection- diffusion differential equations the source term are smooth on the whole domain. Farrell et.al [5] - [6] considered scalar singularly perturbed convection-diffusion equation with discontinuous source term. The interior layers in [6] were strong, in the sense that the solution was bounded but the magnitude of the first derivative grew unboundedly as ε → 0, but in [5] they were weak, in the sense that the solution and the first derivative were bounded but the magnitude of the second derivative grows unboundedly as ε → 0. In this work, we present a uniformly convergent numerical method for a weakly coupled system of singularly perturbed convection-diffusion equations having discontinuous source term with different diffusion parameters. The solution to such equations has overlapping and interacting boundary and interior layers which makes the construction of numerical methods and analysis quite difficult. Tamilselvan and Ramanujam [15] considered the same problem but with equal diffusion parameters.
Consider a weakly coupled system of singularly perturbed convection-diffusion equations with discontinuous source term on the unit interval Ω = (0, 1), having a single discontinuity in the source term at a point d ∈ Ω. Let Ω 1 = (0, d ) and Ω 2 = ( d , 1). Let the jump in a function ω at a point d ∈ Ω given as [ ω ]( d ) = ω ( d +) − ω ( d −). The corresponding boundary value problem is to find u 1 , u 2 C 0 (
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) ∩ C 1 (Ω) ∩ C 2 1 ∪ Ω 2 ), such that
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where E = diag( ε 1 , ε 2 ), the coupling matrix A = diag( a 1 , a 2 ) and B = ( bij ) 2×2 with 0 < ε 1 ε 2 ≤ 1, f = ( f 1 , f 2 ) T , and u = ( u 1 , u 2 ) T . Assume for each i = 1, 2 and x
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, the matrices A and B satisfy
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Let α = min{ α 1 , α 2 }. Further assume that the source terms f 1 , f 2 are sufficiently smooth on
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and their derivatives have jump discontinuity at the same point.
Notations. Throughout the paper, C denotes a generic positive constant and C = ( C , C ) T denotes a generic positive constant vector, both are independent of perturbation parameters ε 1 , ε 2 and the discretization parameter N, but may not be same at each occurrence. Define v w if vi wi , i = 1, 2, and | v | = (| v 1 |, | v 2 |) T . We consider the maximum norm and denote it by║.║ S , where S is a closed and bounded subset in
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. For a real valued function v C ( S ) and for a vector valued function v = ( v 1 , v 2 ) T C ( S ) 2 , we define
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and ║ v S = max{║ v 1 S ,║ v 2 S }. Now let a mesh
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be a set of points satisfying x 0 < x 1 <⋯< xN = 1. A mesh function
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is a real-valued function defined on Ω N . Define the discrete maximum norm for such functions by
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and for vector mesh functions V = ( V 1 , V 2 ) T =
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are used and define║ V ΩN = max{║ V 1 ΩN ,║ V 2 ΩN }.
- 2. The continuous problem
Theorem 2.1 (Continuous maximum principle). Suppose u 1 , u 2 C 0 (
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) ∩ C 1 (Ω) ∩ C 2 1 ∪ Ω 2 ). Further suppose that u = ( u 1 , u 2 ) T satisfies u (0) ≥ 0, u (1) ≥ 0, Lu ( x ) ≥ 0 in Ω 1 ∪ Ω 2 and [ u′ ]( d ) ≤ 0. Then u ( x ) ≥ 0, for all x
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.
Proof . Let u :=
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Now let p and q be the points at which θ 1 ( p ) :=
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and θ 2 ( q ) :=
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. Assume without loss of generality θ 1 ( p ) ≤ θ 2 ( q ). If θ 1 ( p ) ≥ 0, then there is nothing to prove. Suppose that θ 1 ( p ) < 0, then proof is completed by showing that this leads to a contradiction. Note that p ≠ {0, 1}. So either p ∈ Ω 1 ∪ Ω 2 or p = d .
In the first case for x ∈ Ω 2 ,
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In the second case, that is, p = d , we have [ u′ ]( d ) =
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, and at a negative minimum [Θ′]( d ) ≥ 0, which gives a contradiction. □
Lemma 2.2 (Stability Result). Let u = ( u 1 , u 2 ) T be the solution of (1) − (2), then,
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Proof . Define the function Ψ ± ( x ) := max {║ u (0)║,║ u (1)║,║ Lu ║Ω 1 ∪Ω 2 } (2 − x , 2 − x ) T ± u ( x ). Then Ψ ± (0) ≥ 0 , Ψ ± (1) ≥ 0 , L Ψ ± ( x ) ≥ 0 for each x ∈ Ω 1 ∪ Ω 2 , and [Ψ ± ′ ]( d ) = ±[ u′ ]( d ) = 0 , since u C 1 (Ω) 2 . It follows from the maximum principle that Ψ ± ( x ) ≥ 0 for all x ∈ Ω, which leads to the required bound on u . Consequently, the problem (1) − (2) has a unique and stable solution. □
To derive sharper bounds on the derivatives of solution, the solution is decomposed into a sum, composed of a regular component v and a singular component w . That is, u = v + w . The regular component v , can be written in the form v =
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, where v 0 = ( v 01 , v 02 ) T , v 1 = ( v 11 , v 12 ) T and v 2 = ( v 21 , v 22 ) T are defined respectively to be the solutions of the problems
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and
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Thus the regular component v is the solution of
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Further we decompose w as w = w 1 + w 2 where w 1 = ( w 11 , w 12 ) T , w 2 = ( w 21 , w 22 ) T . Thus w 1 = w 11 + w 21 and w 2 = w 12 + w 22 , where w 1 is the solution of
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and w 2 is the solution of
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Lemma 2.3. For each integer k, satisfying 0 ≤ k ≤ 3 , the regular component v and its derivatives satisfy the bounds given by
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Proof . The proof follows from [7] and [2] . □
Lemma 2.4. For each integer k, satisfying 0 ≤ k ≤ 3 , the singular component w 1 and its derivatives satisfy the bounds given by
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Proof . The proof follows from [7] and [2] . □
Lemma 2.5. For each integer k, satisfying 0 ≤ k ≤ 3 , the singular component w2 and its derivatives satisfy the bounds given by
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Proof . Consider the barrier function ϕ (1, 1) T ± w 2 , where
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to bound w 2 . To bound derivatives of w 2 , use the technique used in [7] and bound on w 2 on the domain Ω 1 and Ω 2 . □
3. Discretization of the Problem
We use piecewise uniform Shishkin mesh which uses these transition parameters:
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The interior points of the mesh are denoted by
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Let hi = xi x i −1 be the ith mesh step and
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, clearly
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= { xi : i = 0, 1,..., N }. Let N = 2 l , l ≥ 5 be any positive integer.
We divide
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into three sub-intervals [0, σєl1 ], [ σєl1 , σєl2 ] and [ σєl2 , d ] for some 0 < σєl1 σєl2
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. The sub-intervals [0, σєl1 ] and [ σєl1 , σєl2 ] are divided into N /8 equidistant elements and the sub-interval [ σєl2 , d ] is divided into N /4 equidistant elements. Similarly, in
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the sub-intervals [ d , d + σєr1 ] and [ d + σєr1 , d + σєr2 ] are divided into N /8 equidistant elements and the sub-interval [ d + σєr2 , 1] is divided into N /4 equidistant elements, for some 0 < σєr1 σєr2
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.
Define the discrete finite difference operator LN as follows
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with boundary conditions
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where
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and at x N /2 = d the scheme is given by
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where
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Lemma 3.1. Suppose that a mesh function Z ( xi ) satisfies Z ( x 0 ) ≥ 0,Z ( xN ) ≥ 0, LNZ ( xi ) ≥ 0 for all xi ∈ Ω N and
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0, then Z ( xi ) ≥ 0 for all xi
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.
Lemma 3.2. If Z ( xi ) is any mesh function, then,
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The discrete solution U can be decomposed into the sum U = V + W . The function V , is defined as the solution of the following problem:
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The function W , is defined as the solution of the following problem:
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where the jump in the discrete derivative of a mesh function Z at the point xi = d is given by:
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Further decompose W as W = W 1 + W 2 , where the function W 1 is defined as the solution of the following problem:
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and the function W 2 is defined as the solution of the following problem:
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4. Convergence analysis
By Taylor’s expansion and bounds on regular components defined in lemma 2.3 gives
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Define the mesh function Ψ ± ( xi ) as
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Using discrete maximum principle, the error of the regular component satisfies the estimate
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As in [7] , the error of the singular component satisfies the estimate
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Lemma 4.1. The following ε 1 , ε 2 uniform bound
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where W 2 is the solution of (14) − (15).
Proof . At the point x = d we know that
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First consider
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From lemma 2.3 we have
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Therefore, | D V ( d )| ≤
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Similarly, consider
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Again from lemma 2.3 we have
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Therefore, | D + V ( d )| ≤
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On Ω 1 , | W 1 ( xi )| ≤
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implies that | D W 1 ( d )| ≤
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. On Ω 2 , D + W 1 ( d ) = D + ( W 1 w 1 )( d ) + D + w 1 ( d ). From lemma 2.4
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and | D + ( W 1 w 1 )( xi )| ≤
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.
Therefore,
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Lemma 4.2. The following ε 1 , ε 2 uniform bound
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is valid, where W 2 is the solution of (14) − (15).
Proof . Consider the following function
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= 1, 2 where
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where Ψ = ( ψ 1 , ψ 2 ) T is the solution of
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Using the discrete maximum principle we get the required result.
Lemma 4.3. The error of the singular component satisfies the estimate
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Proof .
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Now
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From lemma 4.1 we have
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and
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Hence
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Likewise,
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Using the bounds on derivatives of w 2 given in lemma 2.5, we have
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Using the bounds on derivatives of w 1 given in lemma 2.4, we have
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Also,
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Collecting all the previous inequalities we get that
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By the Taylor’s expansion and bounds on the derivatives of w 2 , given in lemma 2.5 we have
Case (i) For xi ∈ [ d + σєr2 , 1].
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Similar arguments prove a similar result for the sub-interval [ σєl1 , d ).
Case (ii) For xi ∈ (0, σєl1 ).
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Similar arguments prove a similar result for the sub-interval ( d , d + σєr1 ).
Case (iii) [ σєl1 , σєl2 ).
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Using the inequality
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Likewise, |( LN ( W 2 w 2 )) 2 ( xi )| ≤ CN −1 ln N .
Similar arguments prove a similar result for the sub-interval [ d + σєr1 , d+ σєr2 ).
Combining all these gives,
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Consider the following function
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, j = 1, 2 where
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where Ψ = ( ψ 1 , ψ 2 ) T is the solution of
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Using the discrete maximum principle we get the required result. □
Theorem 4.4. Let u be the solution of given problem (1) − (2) and U is the solution of discrete problem on the Shishkin mesh defined in section 3, then
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Proof . Using the equation (16), (17) and lemma 4.3 we get the required result. □
5. Numerical Results
To illustrate the theoretical results the scheme in Section 3 is implemented on these test examples.
Example 5.1 Consider the following singularly perturbed convection-diffusion problem with discontinuous source term:
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where
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For the construction of piecewise-uniform Shishkin mesh
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, we take α = 0.8. The Exact solution of the examples are not known. Therefore we estimate the error for U by comparing it to the numerical solution
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obtained on the mesh
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that contains the mesh points of the original mesh and their midpoints, that is,
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= xj , j=0,...,N,
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= ( xj + x j +1 )/2, j=0,...,N-1.
For different values of N and ε 1 , ε 2 , we compute
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Maximum point-wise errorsandε1,ε2−uniform rate of convergencepNfor Example 5.1.
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Maximum point-wise errors and ε1, ε2−uniform rate of convergence pN for Example 5.1.
If ε 1 = 10 j for some non-negative integer j , set
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Then the parameter-uniform error is computed as DN :=
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and the order of convergence is calculated using the formula pN :=
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Finally, we want to show that similar results can be obtained for coupled system of M (> 2) singularly perturbed convection diffusion problem with discontinuous source term. Letting N = 2 M × τ , where τ is some positive power of 2, the mesh is defined using the following transition points
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Then we divide the interval [0, d ] into M+1 subintervals [0, σєl1 ], [ σєl1 , σєl2 ],..., [ σєlM−1 , σєlM ], [ σєlM , d ]. On the subinterval [0, σєl1 ] a uniform mesh of N /2 M+1 mesh intervals, on [ σєlk , σєlk+1 ], 1 ≤ k M − 1, a uniform mesh of N /2 M−k+2 mesh intervals, and on [ σєlM , d ] a uniform mesh of N/4 mesh intervals are placed. Similarly, we divide the interval [ d , 1] into subintervals [ d , d + σєr1 ], [ d + σєr1 , d + σєr2 ],..., [ d + σєrM−1 , d + σєrM ], [ d + σєrM , 1]. On the subinterval [ d , d + σєr1 ] a uniform mesh of N /2 M+1 mesh intervals, on [ d + σєrk , d + σєrk +1], 1 ≤ k ≤ M − 1, a uniform mesh of N /2 M−k+2 mesh intervals, and on [ d + σєrM , 1] a uniform mesh of N /4 mesh intervals are placed. Let h єl1 and h єr1 be the mesh lengths on [0, σєl1 ] and on [ d , d + σєr1 ] respectively. Let H 1 and H 2 be the mesh lengths on [ σєlM , d ] and on [ d + σєrM , 1] respectively; hєlk and hєrk be the mesh lengths on [ σєlk , σєlk+1 ] and on [ d + σєrk , d + σєrk+1 ], k = 2,..., M respectively.
In this case also, we obtain the scheme similar to (3.1), with u = ( u 1 ; u 2 ,..., uM ) T
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C 1 (Ω) M C 2 1 ∪ Ω 2 ) M and also expect the error bound
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C ( N −1 ln N ) to hold, although attempts of the authors have failed so far to provide a proof. To illustrate the order of uniform convergence of this method we consider the following test example.
Example 5.2 Consider the following singularly perturbed convection-diffusion problem with discontinuous source term:
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where f 1 ( x ) =
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and
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Maximum point-wise errors, withε2= 10−4,ε3= 10−1andε1,ε2,ε3−uniform rate of convergencepNfor Example 5.2.
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Maximum point-wise errors , with ε2 = 10−4, ε3 = 10−1 and ε1, ε2, ε3−uniform rate of convergence pNfor Example 5.2.
BIO
Sheetal Chawla is doing her Ph.D. at Indian Institute of Technology, Delhi under the direction of Dr. S. Chandra Sekhara Rao. Her area of interests are Numerical Analysis and Differential Equations.
Department of Mathematics, Indian Institute of Technology Delhi, New Delhi-110016.
e-mail: chawlaasheetal@gmail.com
S. Chandra Sekhara Rao is working as a Professor, Department of Mathematics, Indian Institute of Technology, Delhi. His area of interests are Numerical Analysis and Parallel Computing.
Department of Mathematics, Indian Institute of Technology Delhi, New Delhi-110016.
e-mail: scsr@maths.iitd.ac.in
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