# Parameter uniform essentially first order convergence of a fitted mesh method for a class of parabolic singularly perturbed Robin problem for a system of reaction-diffusion equ

In this paper, a class of linear parabolic systems of singularly perturbed second order differential equations of reaction-diffusion type with initial and Robin boundary conditions is considered. The components of the solution u⃗ of this system exhibit parabolic boundary layers with sublayers. A numerical method composed of a classical finite difference scheme on a piecewise uniform Shishkin mesh is suggested. This method is proved to be first order convergent in time and essentially first order convergent in the space variable in the maximum norm uniformly in the perturbation parameters

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09/03/2019

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## 1 Introduction

A differential equation in which small parameters multiply the highest order derivative and some or none of the lower order derivatives is known as a singularly perturbed differential equation. In this paper, a class of linear parabolic singularly perturbed second order differential equation of reaction-diffusion type with initial and Robin boundary conditions is considered.

For a general introduction to parameter-uniform numerical methods for singular perturbation problems, see 11 , 12 , 18 and 19 . In 13 , a Dirichlet boundary value problem for a linear parabolic singularly perturbed differential equation is studied and a numerical method comprising of a standard finite difference operator on a fitted piecewise uniform mesh is considered and it is proved to be uniform with respect to the small parameter in the maximum norm. In 14 , a boundary-value problem for a singularly perturbed parabolic PDE with convection is considered on an interval in the case of the singularly perturbed Robin boundary condition is considered and using a defect correction technique, an ε-uniformly convergent schemes of high-order time-accuracy is constructed. The efficiency of the new defect-correction schemes is confirmed by numerical experiments. In 15 , a one-dimensional steady-state convection dominated convection-diffusion problem with Robin boundary conditions is considered and the numerical solutions obtained using an upwind finite difference scheme on Shishkin meshes are uniformly convergent with respect to the diffusion cofficient.

Consider the following parabolic initial-boundary value problem for a singularly perturbed linear system of second order differential equations

 ∂→u∂t(x,t)−E∂2→u∂x2(x,t)+A(x,t)→u(x,t)=→f(x,t), on Ω, (1)

with

 →u(0,t)−E∗∂→u∂x(0,t)=→ϕL(t),→u(1,t)+E∗∂→u∂x(1,t)=→ϕR(t),0≤t≤T,→u(x,0)=→ϕB(x),0≤x≤1, (2)

where with Here, for all are column vectors, and are matrices, with for all The parameters are assumed to be distinct and for convenience, to have the ordering
The problem (1), (2) can also be written in the operator form

 L→u=→f on Ω,
 β0→u(0,t)=→ϕL(t),β1→u(1,t)=→ϕR(t),→u(x,0)=→ϕB(x),

where the operators are defined by

 L=I∂∂t−E∂2∂x2+A,β0=I−E∗∂∂x,β1=I+E∗∂∂x

where is the identity operator. The reduced problem corresponding to (1), (2) is defined by

 ∂→u0∂t+A→u0=→f, on Ω,→u0=→u on ΓB. (3)

The problem (1), (2) is said to be singularly perturbed in the following sense.
Each component of the solution of (1), (2) is expected to exhibit twin layers of width at and while the components have additional twin layers of width the components have additional twin layers of width and so on.

## 2 Solution of the continuous problem

Standard theoretical results on the existence of the solution of (1), (2) are stated, without proof, in this section. See 16 and 17 for more details. For all it is assumed that the components of satisfy the inequalities

 aii(x,t)>n∑j≠ij=1|aij(x,t)| % for 1≤i≤n, and aij(x,t)≤0 fori≠j (4)

and for some

 0<α

It is also assumed, without loss of generality, that

 √εn≤√α6. (6)

Sufficient conditions for the existence, uniqueness and regularity of a solution of (1), (2) are given in the following theorem.

###### Theorem 2.1

Assume that are sufficiently smooth. Also assume that ,   and the following compatibility conditions are fulfilled at the corners and of

 →ϕB(0)=→ϕL(0)+d→ϕBdx(0)% and→ϕB(1)=→ϕR(0)−d→ϕBdx(1), (7)
 d→ϕLdt(0)=−Ed3→ϕBdx3(0)+Ed2→ϕBdx2(0)+A(0,0)d→ϕBdx(0)−[A(0,0)−∂A∂x(0,0)]→ϕB(0)+→f(0,0)−∂→f∂x(0,0),d→ϕRdt(0)=Ed3→ϕBdx3(1)+Ed2→ϕBdx2(1)−A(1,0)d→ϕBdx(1)−[A(1,0)+∂A∂x(1,0)]→ϕB(1)+→f(1,0)+∂→f∂x(1,0), (8)

and

 d2→ϕLdt2(0)=−E2d5→ϕBdx5(0)+E2d4→ϕBdx4(0)+2EA(0,0)d3→ϕBdx3(0)+[−2EA(0,0)+4E∂A∂x(0,0)]d2→ϕBdx2(0)+[−2E∂A∂x(0,0)+3E∂2A∂x2(0,0)−A2(0,0)+∂A∂t(0,0)]d→ϕBdx(0)+[−E∂2A∂x2(0,0)+A2(0,0)−∂A∂t(0,0)+E∂3A∂x3(0,0)−2A(0,0)∂A∂x(0,0)+∂2A∂x∂t(0,0)]→ϕB(0)+[−A(0,0)+∂A∂x(0,0)]→f(0,0)+∂→f∂t(0,0)−E∂3→f∂x3(0,0)+E∂2→f∂x2(0,0)+A(0,0)∂→f∂x(0,0)−∂2→f∂x∂t(0,0), (9)
 d2→ϕRdt2(0)=E2d5→ϕBdx5(1)+E2d4→ϕBdx4(1)−2EA(1,0)d3→ϕBdx3(1)+[−2EA(1,0)−4E∂A∂x(1,0)]d2→ϕBdx2(1)+[−2E∂A∂x(1,0)−3E∂2A∂x2(1,0)+A2(1,0)−∂A∂t(1,0)]d→ϕBdx(1)+[−E∂2A∂x2(1,0)+A2(1,0)−∂A∂t(1,0)−E∂3A∂x3(1,0)+2A(1,0)∂A∂x(1,0)−∂2A∂x∂t(1,0)]→ϕB(1)+[−A(1,0)−∂A∂x(1,0)]→f(1,0)+∂→f∂t(1,0)+E∂3→f∂x3(1,0)+E∂2→f∂x2(1,0)−A(1,0)∂→f∂x(1,0)+∂2→f∂x∂t(1,0). (10)

Then there exists a unique solution of (1), (2) satisfying .

## 3 Analytical results

The operator satisfies the following maximum principle:

###### Lemma 3.1

Let the assumptions (4) - (6) hold. Let be any vector-valued function in the domain of such that Then    on   implies that   on

###### Lemma 3.2

Let the assumptions (4) - (6) hold. If is any vector-valued function in the domain of then,  for each  and

 |ψi(x,t)|≤max{∥β0→ψ(0,t)∥,∥β1→ψ(1,t)∥,∥→ψ(x,0)∥,1α∥L→ψ∥}.

A standard estimate of the solution

of the problem (1), (2) and its derivatives is contained in the following lemma.

###### Lemma 3.3

Let the assumptions (4) - (6) hold and let be the solution of (1), (2). Then,  for all and each

The Shishkin decomposition of the solution of the problem (1), (2) is

 →u=→v+→w (11)

where and are the smooth and singular components of the solution respectively.
Taking into consideration, the sublayers that appear for the components, the smooth component is subjected to further decomposition.

 vn=u0,n+εnvn,n,vn−1=u0,n−1+εnv1n−1,n,⋮v1=u0,1+εnv11,n, (12)

as all the components have layers. Since components except have sublayers, the components takes the form,

 vn−1=u0,n−1+εn(vn−1,n+εn−1vn−1,n−1),vn−2=u0,n−2+εn(vn−2,n+εn−1v1n−2,n−1),⋮v1=u0,1+εn(v1,n+εn−1v11,n−1). (13)

Further, have sublayers and hence that leads to the decomposition,

 vn−2=u0,n−2+εn(vn−2,n+εn−1(vn−2,n−1+εn−2vn−2,n−2)),vn−3=u0,n−3+εn(vn−3,n+εn−1(vn−3,n−1+εn−2v1n−3,n−2)),⋮v1=u0,1+εn(v1,n+εn−1(v1,n−1+εn−2v11,n−2)). (14)

Proceeding like this, it is not hard to see that

 ⎛⎜ ⎜ ⎜ ⎜⎝v1v2⋮vn⎞⎟ ⎟ ⎟ ⎟⎠=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝u0,1u0,2⋮u0,n⎞⎟ ⎟ ⎟ ⎟ ⎟⎠⎛⎜ ⎜ ⎜ ⎜⎝γ1γ2⋮γn⎞⎟ ⎟ ⎟ ⎟⎠

i.e.

 →v(x,t)=→u0(x,t)+→γ(x,t) (15)

where

 γj=→εn(→vjj)T, (16)
 →εn=(ε1ε2...εn,ε2ε3...εn,......,εn−1εn,εn),→vii=(0,0,...,vi,i,vi,i+1,......,vi,n).

Then using (11) and (15) in (1), (2), it is found that the smooth component of the solution satisfies

 L→v=→f, on Ω (17)

with

 β0→v(0,t)=β0(→u0+→γ)(0,t),β1→v(1,t)=β1(→u0+→γ)(1,t),→v(x,0)=(→u0+→γ)(x,0). (18)

and the singular component of the solution satisfies

 L→w=→0, on Ω (19)

with

 β0→w(0,t)=β0(→u−→v)(0,t),β1→w(1,t)=β1(→u−→v)(1,t),→w(x,0)=→0. (20)

Consider the following parabolic initial-boundary value problem for a singularly perturbed linear system of second order differential equations

 ∂→^u∂t(x,t)−^E∂2→^u∂x2(x,t)+^A(x,t)→^u(x,t)=→^f(x,t), on Ω, (21)

with

 ^u2(0,t)−√εn∂^un∂x(0,t)=α(t),^u2(1,t)+√εn∂^un∂x(1,t)=β(t),0≤t≤T,→^u(x,0)=→δ(x),0≤x≤1, (22)

where is a matrix, with
The problem (21), (22) can also be written in the operator form

 ^L→^u=→^f on Ω,
 b0^un(0,t)=α(t),b1^un(1,t)=β(t),→^u(x,0)=→δ(x).

where the operators are defined by

 ^L=I∂∂t−^E∂2∂x2+^A,b0=I−√εn∂∂x,b1=I+√εn∂∂x

where is the identity operator. The reduced problem corresponding to (21), (22) is defined by

 ∂→^u0∂t+^A→^u0=→^f, on Ω,→^u0=→^u on ΓB.

The operator satisfies the following maximum principle:

###### Lemma 3.4

Let the assumptions (4) - (6) hold. Let be any vector-valued function in the domain of such that Then    on   implies that   on

###### Lemma 3.5

Let the assumptions (4) - (6) hold. If is any vector-valued function in the domain of then,  for each and

A standard estimate of the solution of the problem (21), (22) and its derivatives is contained in the following lemma.

###### Lemma 3.6

Let the assumptions (4) - (6) hold and let be the solution of (21), (22). Then,  for all and each

Bounds on the smooth component of and its derivatives are contained in

###### Lemma 3.7

Let the assumptions (4) - (6) hold. Then there exists a constant such that, for each and ,

 |∂lvi∂tl(x,t)|≤C,l=0,1,2,|∂lvi∂xl(x,t)|≤C,l=1,2,|∂lvi∂xl(x,t)|≤Cε−(l−2)/2i,l=3,4,|∂l+1vi∂xl∂t(x,t)|≤C,l=1,2.

Proof. From (12) - (14) it is observed that the components satisfy the following systems of equations:

 ∂v1,n∂t+a11v1,n+a12v2,n+...+a1nvn,n=ε1εn∂2u0,1∂x2∂v2,n∂t+a21v1,n+a22v2,n+...+a2nvn,n=ε2εn∂2u0,2∂x2⋮∂vn−1,n∂t