UNIFORMLY CONVERGENT 3-TGFEM VS LSFEM FOR SINGULARLY PERTURBED CONVECTION-DIFFUSION PROBLEMS ON A SHISHKIN BASED LOGARITHMIC MESH

INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING, SERIES B Volume 4, Number 4, Pages 35 334 c 23 Institute for Scientific Computing and Information UNIFORMLY CONVERGENT 3-TGFEM VS LSFEM FOR SINGULARLY PERTURBED CONVECTION-DIFFUSION PROBLEMS ON A SHISHKIN BASED LOGARITHMIC MESH VIVEK SANGWAN AND B. V. RATHISH KUMAR Abstract. In the present work, three-step Taylor Galerkin finite element method(3tgfem) and least-squares finite element method(lsfem) have been discussed for solving parabolic singularly perturbed problems. For singularly perturbed problems, a small parameter called singular perturbation parameter is multiplied with the highest order derivative term. As this singular perturbation parameter approaches to zero, a very sharp change occurs in the solution, which makes it difficult to find solution by traditional methods unless some special treatment is employed. A comparison on the performance of the three schemes namely, (a) 3TGFEM with exponentially fitted splines, (b) explicit least-squares finite element method with linear basis functions and (c) 3TGFEM with linear basis functions, for solving the parabolic singularly perturbed problems has been made. For all the three schemes Shishkin based logarithmic mesh has been used for numerical computations. It has been found out that the 3TGFEM scheme with exponentially fitted splines provides more accurate results as compared to the other two schemes. Detailed error estimates for the three-step Taylor Galerkin scheme with exponentially fitted splines have been presented. The scheme is shown to be conditionally uniform convergent. It is third order accurate in time variable and linear in space variable. Numerical results have been presented for all the three schemes for both linear and non-linear problems. Key words. Boundary layer, singularly perturbed problems, mass-lumped schemes, finite element method, Taylor Galerkin method, exponentially fitted splines, least-squares method, error estimates, uniform convergence.. Introduction Singularly perturbed problems appear in many areas of science and technology such as hydroelectric theory, chemical reactor theory, aerodynamics, fluid mechanics, heat transfer and problems in structural mechanics [,2] etc. where the diffusion co-efficient can be very small as compared to the convection co-efficient. When this diffusion co-efficient becomes smaller and smaller as compared to the convection co-efficient, sharp boundary layers evolve in the solution. Because of these boundary layers, conventional methods fail to approximate the solution, particularly in the layer region. A lot of work has been done by Martin Stynes et al. [2] in this regard. We need to use very robust numerical schemes for solving the Singularly Perturbed Problems(SPP). Three-step Taylor Galerkin finite element(3tgfe) scheme is one such robust higher order numerical scheme for solving the convection dominated problems. J. Donea [3] was the first to propose the scheme for convection dominated flow problems. Because of its inherent upwinding, the scheme has been successfully employed for solving highly convective transport problems by Donea et al. [3, 4]. Later Jiang and Kawahara used the method for solving the Navier-Stokes equation [8]. As demonstrated by Peter D. Lax [5 7], performing time discretization prior to spatial discretization leads to better time accurate schemes with improved stability properties, as compared to conventional methods, 3TGFE method enjoys this benefit. The method produces particularly high phase Received by the editors April, 23, and, in revised form, September 3, 23. 2 Mathematics Subject Classification. 35R35, 49J4, 6G4. 35

36 V. SANGWAN AND B. KUMAR accuracy and improved stability properties [4]. In this study we propose 3TGFEM for solving parabolic singularly perturbed problems. In addition to this, we have used Shishkin based logarithmic mesh in order to capture the boundary layer more sharply in the boundary layer region. Since the Bubnov Galerkin method does not provide accurate computational solution and as is suggested by Roos and Martin Stynes et al. [2,8,9] to use the exponentially fitted splines for the method to be u- niformly convergent, we have also used exponentially fitted splines for the proposed 3TGFE scheme to get the added advantages of the exponentially fitted splines. Least-squares finite element method(lsfem) is also known for its better inherent upwinding properties. For the solution of problems in convection dominated and high-speed compressible flows, the LSFEM inherently contains a mechanism to automatically capture discontinuities, shocks or boundary layers [9]. Because of these properties and the universality, efficiency and robustness of the LSFEM, the method has been widely used for solving convection dominated flow problems, hyperbolic problems, Navier-Stokes equations and many more problems arising in the field of science and engineering [9 ]. But only a very few researchers have used LSFEM for solving SPP. Evrenosoglu and Somali approximates the solution of singularly perturbed two-point boundary value problems with LSFEM using Bezier control points[]. In this study we also propose LSFEM for singularly perturbed parabolic differential equations. The robustness, efficiency and accuracy of the proposed three schemes, namely, (a) 3TGFEM with exponentially fitted spline basis, (b) explicit LSFEM with linearly fitted spline basis and (c) 3TGFEM with linearly fitted spline basis have been tested both on linear and nonlinear problems using Shishkin based logarithmic mesh partitioning. Based on these numerical experiments, it has been observed that 3TGFEM with exponentially fitted splines produces more accurate results as compared to the other two schemes. Error estimates have been derived for the proposed 3TGFE scheme with exponentially fitted splines. The organization of the paper is as follows. In Section 2, the continuous problem has been defined. Section 3 contains some results on bounds of the exact solution and its derivatives. Section 4 deals with the 3TGFEM, explicit least-squares finite element formulation and the logarithmic mesh. In Section 5, the uniform convergence of the proposed scheme has been derived and in the last Section, numerical results have been presented both for linear and nonlinear problems. 2. The Continuous Problem under Consideration We consider the following time-dependent singularly perturbed convection-diffusion problem (2.) L ϵ u(x, t) u t ϵu xx + b(x)u x = f(x, t), (x, t) Ω with initial condition (2.2a) and boundary conditions as (2.2b) (2.2c) u(x, ) = u (x) for x [, ] u(, t) = g (t) for t [, ], u(, t) = g (t) for t [, ], where Ω = (, ) 2 and the singular perturbation parameter satisfies < ϵ. We assume that the convection co-efficient b(x) and the source term f(x, t) are

3-TGFEM VS LSFEM FOR CONVECTION-DIFFUSION PROBLEMS 37 sufficiently smooth functions with (2.3) < β b(x) β on Ω. Since, we have assumed b(x) on [, ], the boundary layer will appear in the outflow boundary x = [3,4]. Also, Roos et al. [2] has shown the uniqueness of the solution for problems (2.) and (2.2) provided the data is smooth and compatible. We assume the compatibility conditions as (2.4) u () = g () and u () = g (), so that the data matches at the end points (,) and (,) of the domain Ω. Also, we assume that the initial solution u is smooth enough to ensure the continuity and ϵ-uniform bound for the solution of problem (2.). This regularity will be required to obtain the appropriate space and time accuracy. Lemma 2. (Bobisud [6]): Let < ϵ. Then, under the compatibility conditions defined by (2.4), the solution u(x, t) of the continuous problem (2.) satisfies (2.5) (2.6) u(x, t) u (x) Ct (x, t) Ω and u(x, t) g (t) Cx (x, t) Ω. for some constant C. The reduced problem can be obtained by taking the singular perturbation parameter ϵ equal to zero in Eq.(2.). (2.7a) (2.7b) (2.7c) u r t + b(x)u r x = f(x, t) (x, t) Ω u r (x, ) = u (x) for x, u r (, t) = g (x) for t. Now, as ϵ, the solution u(x, t) of the original problem (2.) will converge to the solution of the reduced problem (2.7) away from the boundary layer side x =. It is clear that to obtain the error bounds on the solution of the continuous problem, it is assumed that the solution of the reduced problem (2.7) is sufficiently smooth. 3. Some results on the asymptotic behavior of the exact solution: In the analysis of the uniform convergence of the proposed scheme, we also need to know the asymptotic behavior of the exact solution u(x, t) of Eq.(2.) and its derivatives with respect to the singular perturbation parameter ϵ. The asymptotic behavior of the solution and its derivatives is given by the following results to follow. We assume that u(x, t) is sufficiently smooth to allow both differentiations of (2.) on Ω and the interchange of the order of differentiation in mixed derivatives. Lemma 3. (Bobisud [6]): The exact solution u(x, t) of the problem (2.) can be written as u(x, t) = u (x, t) + ϵu 2 (x, t) + P (x, t), where u and u 2 satisfies parabolic equations similar to (2.) with zero initial-boundary conditions and P is independent of ϵ. From the lemma, it is clear that we may assume, without any loss of generality, zero initial-boundary conditions for the purpose of bounding the derivatives of the exact solution of the problem (2.). Lemma 3.2: The solution u(x, t) is bounded on Ω, i.e. (3.) u(x, t) C (x, t) Ω.

38 V. SANGWAN AND B. KUMAR Proof: The result follows from the inequality (2.5). Lemma 3.3: The differential operator L ϵ defined by Eq.(2.) satisfies a maximum principle. Proof: Suppose there exists some function v C 2, (Ω) for which there exists some point (x, t ) Ω such that v(x, t ) = min v(x, t) <. (x,t) Ω By hypothesis of maximum principle, (x, t ) / Ω (x, t ) Ω. Since (x, t ) is a point of minima for v, we have v xx (x, t ), v x (x, t ) =, v t (x, t ) =. Applying the differential operator L ϵ on v gives L ϵ v(x, t ) = v t (x, t ) ϵv xx (x, t ) + b(x, t )v x (x, t ) L ϵ v(x, t ) <, a contradiction to the fact that L ϵ v(x, t) on Ω and this proves the result. Lemma 3.4: u t (x, t) C on Ω. Proof: On the boundaries x = and x = of the spatial domain [, ], we have u = (Using Lemma 3.) u t = Again, on the initial temporal boundary, by Lemma (3.), we have u = Therefore, Eq.(2.) gives u x = and u xx =. u t (x, ) = f(x, ) for x. Now, using the smoothness condition of f(x, t), we can find some constant C, large enough such that u t C on the boundary Ω. Considering the same operator L ϵ defined by Eq.(2.) and applying on u t, we have L ϵ (u t )(x, t) = u tt ϵu txx + bu tx = (u t ϵu xx + bu x ) t = f t Again, using the smoothness properties of f, we can have L ϵ u t (x, t ) C 2 for some large constantc 2. Since, by Lemma 3.3, the operator L ϵ satisfies the maximum principle, we can have u t C on Ω, for some suitably chosen constant C. Lemma 3.5: Prove that u x (x, t) C( + ϵ e β( x) ϵ ) (x, t) Ω.

3-TGFEM VS LSFEM FOR CONVECTION-DIFFUSION PROBLEMS 39 Proof: For a fix t [, ], the result follows by applying the arguments of Kellogg and Tsan [7] on the line segment {(x, t) : x } as the only bounds needed are u C and u t C on Ω. We conclude the following result from Lemma (3.)- (3.5): Theorem 3.6: The exact solution u(x, t) of the problem (2.) satisfies the bounds (3.2) ( x ) i ( t ) j ( u(x, t) C ) + ϵ i e β( x) ϵ (x, t) Ω, where i and j are non-negative integers such that i, j and i + j. 4. Three-step Taylor Galrkin Finite Element Method: Multi-step Taylor Galerkin schemes were proposed by J. Donea and were based on the Lax-Wendroff finite difference approach. In Taylor Galerkin schemes, the time discretization is carried out using higher order Taylor series expansion. Then this higher order approximation is divided into multi-steps in order to reduce the higher order smoothness and higher order approximation functions requirements. After this temporal discretization, spatial discretization is performed using Galerkin finite element formulation. This very feature of temporal discretization before spatial discretization empowers the Taylor Galerkin finite element schemes to sustain its higher-order spatial accuracy which is generally lost due to the reverse order of approximation [5 7]. We will describe in brief the 3TGFEM by applying it on heat equation (4.) u t = νu xx for ν > and t. The Taylor series expansion up to third order is given by (4.2) u n+ = u n + tu n t + t2 2 un tt + t3 6 un ttt + O( t 4 ), where u n stands for u(x, t n ) and t = t n+ t n. Differentiating successively E- q.(4.) with respect to time, we get (4.3) u tt = ν 2 u xxxx, u ttt = ν 3 u xxxxxx. Substituting (4.3) into the Taylor series expansion (4.2), the higher order semidiscrete formulation in time becomes, (4.4) u n+ = u n + tνu xx + t2 2 ν2 u xxxx + t3 6 ν3 u xxxxxx + O( t 4 ) One can clearly note that one needs to choose very high order approximation functions for the purpose of spatial discretization of (4.4), whereas the three-step Taylor Galerkin approach for (4.2) leads to (4.5) u n+/3 = u n + t 3 un t, u n+/2 = u n + t 2 un+/3 t, u n+ = u n + tu n+/2 t. From this three-step Taylor Galerkin formulation, one can observe that the compulsion of choosing the higher-order approximating functions is reduced and it is sufficient to take linear approximation for spatial discretization while retaining the

32 V. SANGWAN AND B. KUMAR third order temporal accuracy. Also, for a given physical problem to handle the higher order derivatives is not an easy task. Three-step Taylor Galerkin schemes overcome these problems of higher order smoothness requirements and the calculations of higher order derivatives in a very natural way. For spatial discretization, we proceed as follows: We define the solution space as and test space as S = {u (H (, ); L 2 (, ))}, T = {w (H (, ); L 2 (, )) : w(, t) =, w(, t) =, ϵw + b(x)w = }. Now, the variational formulation of the three-step Taylor Galerkin scheme for the continuous problem (2.) can be obtained by multiplying (4.5) with a test function v T and performing integration by parts. To obtain a fully discrete equation, we apply the standard Petrov Galerkin finite element method. The spatial domain discretization is explained in [5]. Let k = M be the time spacing parameter and t m = k m for m =,, 2, 3,...M. Let Ω h,k denotes the discretization of Ω at k-th time level. Let Ω e denotes a linear element (x e, x e+ ). The finite dimensional subspaces S h and T h of the solution space S and the test space T are defined by S h = span{ϕ i (x)} N i= and T h = span{ψ i (x, t)} N i=, where {ϕ i (x)} and {ψ i (x, t)} are the sets of approximation functions and test functions respectively. Now, the discrete approximation to the exact solution y(x, t) is given by y(x, t) = N y i (t)ϕ i (x), where {y i } are nodal values. i= On substituting the above finite element level discretization into the variational formulation, we get a corresponding system of linear equations which can be written as [A]{y} = {f} Solving the above system for the nodal values {y i } gives rise to the 3TGFE solution. 4.. Mesh selection strategy: In the process of finite element discretization of the given problem, the spatial domain is discretized using logarithmic mesh based on Shishkin s approach. As ϵ, the boundary layer evolves at x =, therefore, higher density of nodal points is expected at the boundary x =. Based on this idea, Shishkin proposed piecewise uniform Shishkin mesh which is frequently used for domain discretization for solving SPP. For constructing the logarithmic mesh, we firstly discretize the spatial domain into piecewise uniform Shishkin mesh. The boundary layer thickness σ x is chosen by σ x = min{.5, Kϵ ln(n)} where K is a constant to be chosen such that K β and N( 4) is the number of mesh elements taken in the x-direction. Therefore, the piecewise uniform mesh for Ω N σ x is given by { { } 2( σx ) Ω N σ x = x i : x i = N i for i N 2 ( σ x ) + 2σx N (i N 2 ) for N 2 < i N Again, these last N 2 mesh points lying in the boundary layer region are mapped to new mesh points using a logarithmic function in such a way that the new mesh points will be condensed near the boundary x =. This condensing of mesh points

3-TGFEM VS LSFEM FOR CONVECTION-DIFFUSION PROBLEMS 32 in the boundary layer region helps the numerical scheme in capturing the boundary layers more sharply and also in improving the accuracy and stability properties. Another important aspect of this kind of logarithmic mesh is the smoothness property of the mesh points which means that the mesh becomes finer and finer in a smooth way rather than becoming fine all of a sudden. 4.2. Explicit Least-Squares Finite Element Method: The least-squares finite element method(lsfem) is based on the minimization of the residuals in a least-squares sense over the computational domain. A brief outline of the leastsquares finite element formulation for the given problem (2.) is given as follows: With the introduction of a new variable u x = p, the singularly perturbed boundary value problem (2.) is equivalent to the firstorder system ( ) ( ) ux p (4.6) =, < x < ; ϵp x u t b(x)u x + f(x, t) (4.7) with ( ) ( ) u(, t) p(, t) + ( ) ( ) u(, t) p(, t) = ( ) g (t), < t < ; g (t) and u(x, ) = u (x). Using the standard notations from the Sobolev space theory, we introduce the spaces V = {v(x, t) : v (H (Ω); L 2 (Ω))} and W = {w(x, t) : w (H (Ω); H (Ω))}. The least-squares formulation for (4.6)-(4.7) seeks (u, p) V W such that the following quadratic functional is minimized: (4.8) I(u, p) = ( ut ϵp x + b(x)u x f(x, t) 2,Ω + u x + p 2 ),Ω ds We consider the uniform subdivision of the temporal domain { = t < t < t 2 <... < t M = } of [, ] with step-size τ = M. We consider the one-step methods which consists of stepping only from one timelevel t to t + τ for minimizing the least-square functional (4.8). Using the forward- Euler method for discretizing time derivative in (4.8), we get I(u, p) u(x, t + τ) u(x, t) ( ϵp x (x, t) + b(x)u x (x, t) f(x, t) 2,Ω τ τ + u x (x, t + τ) + p(x, t + τ) 2,Ω)ds Applying the minimization method leads to the time-discretized least-squares formulation of the problem (2.). Now the spatial discretization is performed in the similar way as in the three-step Taylor Galerkin finite element formulation. Here, in the finite element formulation, we have used the piecewise linear basis functions.

322 V. SANGWAN AND B. KUMAR 5. Uniform Convergence for the Three-step Taylor Galerkin Finite Element Method with Exponentially Fitted Splines: In the current section, the uniform convergence in the pointwise maximum norm for the proposed 3TGFE method with exponentially fitted splines has been shown. In this process firstly the stability estimates have been derived and then the scheme is shown to be conditionally uniform convergent and is of first order accurate in space variable and third order accurate in time variable. We start with defining the approximation b(x) of b(x) by (5.) b(x) = b(xi ) for x (x i, x i ] for i =, 2, 3,..., N. For a fixed time level t j, we define the trial and test functions on each of the line segments [, ] {t j } as follows: Choose the trial functions {ϕ i (x)} N i= to be the standard piecewise linear hat functions which are independent of time variable t, i.e. N ϕ i xx = on (x i, x i ), (5.2) i= and ϕ i (x j ) = δ ij j =,, 2,..., N, where δ ij is the Kronecker delta. Next, we choose the test functions {ψ i (x)} N i= as exponentially fitted L splines to be the solutions of ϵψxx(x) i b(x)ψ N x(x) i = on (x i, x i ), (5.3) i= and ψ i (x j ) = δ ij for j =,, 2,..., N, where each ψ i is compactly supported with support (x i, x i+ ). Here, note that the test functions ψ i (x, t) are functions of both space and time variable, but as mentioned earlier that the functions are defined at time level t = t j, the variable t has been omitted from ψ i (x, t). We define the approximation (5.4) y t (x, t)ψ j (x)dx h j y t (x j, t). The schemes using this kind of approximation are called the mass-lumped schemes. We define the coefficients of the problem (2.) at nodal points (x i, t m ) by y i,m = y(x i, t m ), b i = b(x i ) f i,m = f(x i, t m ). Now, we evaluate the bilinear form B m (y, ψ i ) which appears on the R.H.S. of (4.5) at the time of variational formulation in the 3TGFE scheme. We have B m (w, v) = (ϵw x v x + b(x)w x v) m dx, where subscript m tells that the computation is being carried out at t = t m. Using integration by parts and using Eq.(5.2) and (5.3), we get B m (ϕ i, ψ i ) = ϵϕ i (x)ψ i x(x) x i x i = ϵψ i x(x + i )

3-TGFEM VS LSFEM FOR CONVECTION-DIFFUSION PROBLEMS 323 Similar expressions can be obtained for B m (ϕ i, ψ i ) and B m (ϕ i+, ψ i ). Therefore, B m (y, ψ i ) = i+ j=i y j,m Bm (ϕ j, ψ i ) [ h i ρ i = ϵ e ρ (y i i,m y i,m ) h i+ ρ ] i+ e ρ (y i+ i,m y i+,m ) B m (y, ψ i ) = ϵ [ h i σ( ρ i )(y i,m y i,m ) h i+ σ(ρ i+)(y i,m y i+,m ) ] where ρ i = bihi ϵ and σ(x) = x e x. The third order Taylor Galerkin approach is given by (5.5) Bm (y, ψ i ) + h i ( yi,m+ y i,m k k ) 2 (y tt) i,m k2 6 (y ttt) i,m = h i f i,m for i =, 2, 3,..., N and m =,, 2,...M. Using (4.5) approach, this Eq.(5.5) can be written in three steps to find the solution at (m + )-th time level. A general form of each of these three equations can be written in the form as ϵ [ h i σ( ρ i )(y i,m y i,m ) h i+ σ(ρ i+)(y i,m y i+,m ) ] (5.6) +h i ( yi,m+ y i,m k We can write the scheme (5.6) in the matrix form as ) = h i f i,m. (5.7) (A m + k Ĩ) y m + (k Ĩ + J) y m+ = f m + g m+, where (A m ) (N+) (N+) is a tridiagonal matrix with th and Nth rows identically zero and Ĩ(N+) (N+) is the identity matrix with first and last rows with zero entries. For i =, 2, 3,..., N, we have (A m ) i,i = ϵh 2 i σ( ρ i ), (A m ) i,i+ = ϵh i h i+ σ(ρ i+), (A m ) i,i = (A m ) i,i (A m ) i,i+ ; m =,, 2,..., M. Matrix (J) (N+) (N+) is used to incorporate the boundary conditions with (, ) and (N, N) entries equal to and all other entries zero. Also, we have used the notations y m = (y, y, y 2,..., y N ) T m, f m = (, f, f 2,..., f N, ) T m, g m = (g (t m ),,,...,, g (t m )) T, where T denotes the transpose. Since k Ĩ + J is a diagonal matrix with positive entries and hence is invertible. Therefore, from Eq.(5.7), we have (5.8) y m+ = (k Ĩ + J) [(A m + k Ĩ) y m + f m + ] g m+.

324 V. SANGWAN AND B. KUMAR Next, we will find conditions on h i and k for which all the entries of the tridiagonal matrix [A m + k Ĩ] (N+) (N+) are positive. Choosing gives (A m ) i,i = h i b i, e b i h i ϵ (A m ) i,i+ = h i+ b i+, e b i+ h i+ ϵ k(a m ) i,i = k[(a m ) i,i + (A m ) i,i+ ]. h = k(a m ) i,i kh [ min h i, i N β e b i h ϵ + β e b i+ h ϵ Since e and x e x are decreasing functions for all x, we can always choose h and k such that ( ) (5.9) kh β β + < e b i h ϵ e b i+ h ϵ (ka m + Ĩ) i,i >. We define the discrete operator L h,k as (5.) (L h,k y) i,m B m (y m, ψ i ) + h i ( yi,m+ y i,m k ]. k ) 2 (y tt) i,m k2 6 (y ttt) i,m. Since by assumption, we have y, the following theorem is a straight forward consequence of (5.8) and (5.9). Please note that the above steps can be carried out independently for each of the three steps involved in the multi-time step scheme, which in turn will lead to three potential values for time increment k. One can proceed further with the choice of k to be the minimum of these three values. Theorem 5.: Consider the operator L h,k defined by (5.). If the mesh widths h and k are so chosen that the inequality (5.9) is satisfied, then the discrete operator L h,k satisfies a discrete maximum principle. Lemma 5.2: For a fixed t [, ], the following inequality holds (5.) (, ψ i ) h i Ch 2 i. Proof: Since we have graded logarithmic mesh such that h j h k for j < k where h j = x j x j, with a simple calculation, we can get where g(z) = (e z ) z, p = bihi ϵ (5.2) (, ψ i ) h i h i g(q) g(p) (, ψ i ) h i [ + g(p) g(q)], and q = bi+hi ϵ. h i q p. g (ξ i ) (using Mean Value Theorem). Again, with a simple calculation, we can get g ( ) C and g (z) C z 2 for z. Now, the result follows from the following two cases:

3-TGFEM VS LSFEM FOR CONVECTION-DIFFUSION PROBLEMS 325 Case(i): h i ϵ From (5.2), we have Case(ii): h i ϵ Again from (5.2), we have (, ψ i ) h i Ch i b i+ b i h i ϵ Ch 2 i. (, ψ i ) h i Ch i b i+ b i Ch 2 i. ( ) ( ) 2 hi ϵ ϵ βh i Theorem 5.3: Assume that the coefficients of the problem (2.) are smooth enough so that the estimates (3.2) holds and the stability condition (5.9) is satisfied, then u(x i, t m ) y i,m C(h i + k 3 ) i, m. Proof: We shall prove the result for a fixed i {, 2, 3,..., N } and m {, 2, 3,..., M }. Let e i,m = u(x i, t m ) y i,m. We use the notation O(h p k q ), where p and q are integers, to denote any quantity bounded in absolute value by Ch p k q. Consider the operator (L h,k v) i,m = h i h ik 2 6 (v ttt) i,m }, where B(u, v) i,m = (ϵu x, v x ) i,m + ( bu x, v) i,m. { B(v, ψ i ) i,m + h i ( v i,m+ v i,m ) h ik k 2 (v tt) i,m h i (L h,k e) i,m = B(u, ψ i ) i,m + h i ( u i,m+ u i,m ) h ik k 2 (u tt) i,m h ik 2 6 (u ttt) i,m h i f i,m = {(( b b)u x, ψ i ) i,m + (f, ψ i ) i,m h i f i,m } (u t, ψ i ) i,m + h i (u t ) i,m + O(h i k 3 ) {(( b b)u x + f m f i,m u t + (u t ) i,m, ψ i ) i,m } + O(h 2 i ) + O(h i k 3 ) (using Lemma (5.2)). Using Theorem 3.6, the above inequality can be written as (L h,k e) i,m C xi+ x i A simple integration reduces this inequality to ϵ e β( x) ϵ dx + Ch i + Ck 3. (L h,k e) i,m Ce β( x i+ ) ϵ ( e 2ρi ) + Ch i + Ck 3 ; ρ i = βh i ϵ. Choose the barrier function w = Ch i ( + e ρ i 2 ) 2 e β( x) 2ϵ + C(h i + k 3 )x on Ω.

326 V. SANGWAN AND B. KUMAR On boundary points x =, x = and t =, it is clear that e w. Again, operating L h,k on w gives (L h,k w) i,m = h i B(w, ψ i ) i,m xi+ h ( ϵw xx (x) + b m w x (x))ψ i,m (x)dx. x i Since ϵw xx (x) + b m w x (x) Cβh i ϵ ( + e ρ i 2 ) 2 e β( x) 2ϵ + C(h i + k 3 ) (L h,k w) i,m xi+ x i (Cϵ ( + e ρi 2 ) 2 e β( x) 2ϵ + C(h i + k 3 ))ψ i,m dx Using the grading property of the logarithmic mesh, we get (L h,k w) i,m C( e 2ρ i )e β( x i+ ) ϵ + C(h i + k 3 ) (L h,k w) i,m (L h,k e) i,m. Since, the operator L h,k satisfies the discrete maximum principle, we have Hence, we get e i,m w i,m i.e. u(x i, t m ) y i,m Ch i ( + e ρ i 2 ) 2 e β( x) 2ϵ + C(h i + k 3 )x 6. Numerical Results: Ch i + C(h i + k 3 ) = Ch i + Ck 3. u(x i, t m ) y i,m Ch i + Ck 3 i, m. In this section, we will present some numerical results for both the linear and nonlinear problems to support the theoretical results presented in the earlier sections. All the three proposed schemes have been used to solve the one-dimensional singularly perturbed partial differential equations(sppde) numerically. We observed that the scheme (a) 3TGFEM with exponentially fitted splines provides more accurate results as compared to the other two schemes namely (b) explicit LSFEM with linear splines and (c) 3TGFEM with linear splines. In all the cases, time-step has been chosen to satisfy the stability conditions. Example 6.: Consider the linear SPPDE with initial condition (6.a) u t (x, t) ϵu xx (x, t) u x (x, t) = f(x, t) on Ω = [, ] 2, and boundary conditions as (6.b) (6.c) u(x, ) = for x [, ], u(, t) = sin(2t) for t [, ], u(, t) = for t [, ]. The source term f(x, t) is so chosen to satisfy the exact solution x e( ϵ ) e ( ϵ ) ( πx u(x, t) = 2 e ( ϵ ) sin(2t) + 2xcos ) sin(t) for x and t.

3-TGFEM VS LSFEM FOR CONVECTION-DIFFUSION PROBLEMS 327 As we know the exact solution, we compute the pointwise errors by e N,δt ϵ (i, j) = u(x i, t j ) y i,j, where the superscript N indicates the number of mesh elements used in the spatial direction and y i,j is the computed solution at the nodal point x i and at time level t j. Note that we have used the Shishkin based logarithmic mesh, therefore, not all x i are uniformly placed. For each ϵ, we define the maximum nodal error as E ϵ,n,δt = max i,j e N,δt ϵ (i, j). Now, for each N and each time level δt, the ϵ-uniform maximum nodal error is defined by E N,δt = max E ϵ,n,δt. ϵ Proceeding as in [2], the numerical rate of convergence is given by p ϵ,n = log(e ϵ,n,δt/e ϵ,2n, δt 2 log 2 and the numerical ϵ-uniform rate of convergence is given by p N = log(e N,δt/E 2N, δt 2 log 2 For all the three schemes, all these maximum nodal errors and numerical rate of convergence are presented in table, table 2 and table 3 respectively. A comparison of these tables clearly shows that the proposed 3TGFEM with exponentially fitted splines provides a more powerful tool than the other two schemes for solving the SPPDE. The scheme produces more accurate results than the other two schemes. Here, we would like to mention that the rate of convergence presented in all the three tables are approximate results. Since the logarithmic mesh based on Shishkin mesh selection strategy has been used and the transition parameter will change as we move from N = 64 to N = 28 and from N = 28 to N = 256 etc making it is very difficult to use the numerical rate of convergence formula accurately as proposed above. ). ), Now, we plot the computational solution in Fig. and Fig. 2 to explore the boundary layer behavior and for proposing the grids for all the three schemes for the given problem. In Fig. (a-c), the computed solution has been plotted for different grids for all the three proposed schemes, for different values of the singular perturbation parameter ϵ. If we observe the solution plots and the maximum error tables more carefully, we notice that for scheme (a), a grid of 64 elements is adequate enough to capture very sharp boundary layers. Though a grid of 64 elements seems to be sufficient enough for the other two schemes, a grid of 28 elements is proposed for these schemes to capture the boundary layers more sharply. The grid validation check has been carried out for a large set of parameters and as a sample, only one plot for each of the schemes has been presented. We suggest the reader to look upon the maximum error tables for more details. Also these plots depicts that all the three proposed schemes are very much efficient in capturing very sharp boundary layers.

328 V. SANGWAN AND B. KUMAR Table. Numerical maximum errors (E N ϵ ) and numerical rate of convergence (p N ) for 3TGFEM with exponential fitted splines: ϵ N = 32 N = 64 N = 28 N = 256 2.9E-4 24E-4.563E-5.44E-5-2.8.32.36 2 2.5E-3.373E-4.937E-5 37E-5 3.3 2.99. 2 3 36E-3.59E-4.48E-4.37E-5 3.27 3.9 2 4.496E-3 5E-3.33E-4.782E-5.38 4.6.42 2 5.986E-3 55E-3 36E-4.59E-4.35 4.26 -.5 2 6.986E-3 83E-3 83E-4 66E-4.25 3.53 3 2 7.8E-2 83E-3 83E-4 66E-4.34 3.53 3 2 8.39E-2.33E-3 83E-4 66E-4.49 3.63 3 2 9.55E-2.376E-3 83E-4 66E-4.42 3.8 3 2.55E-2.45E-3.988E-4 66E-4.34.4.3 2.55E-2.45E-3.4E-3 66E-4.34.36.36 2 2.55E-2.45E-3.4E-3 66E-4.34.36.36 2 3.55E-2.45E-3.4E-3 66E-4.34.36.36 2 4.55E-2.45E-3.4E-3 66E-4.34.36.36 2 5.55E-2.45E-3.4E-3 66E-4.34.36.36 E N,δt.55E-2.45E-3.4E-3 66E-4 p N.34.36.36 In plots (d-f), a comparison of the computed solutions from all the three schemes has been made with the exact solution of the problem for different sets of values of the singular perturbation parameter ϵ. It can be easily noticed that over the proposed grids for these schemes, all the schemes are in very much agreement with the exact solution. This validates both the schemes and the codes. We can also notice the evolution of boundary layers as the singular perturbation parameter approaches to. Also one can observe that the boundary layer effect becomes very negligible for smaller values of ϵ after ϵ = 2 5. Therefore, we have considered the singular perturbation parameter values up to 2 5. Since all the solution profiles for all the three schemes are same, for the rest of the solution plots, we restrict ourselves to the 3TGFEM with exponentially fitted splines. Fig. 2(a-b) depicts the evolution of the boundary layers for ϵ = 2 5 and 2 at different time levels. From plot 2(b), we can notice that the schemes are very much

3-TGFEM VS LSFEM FOR CONVECTION-DIFFUSION PROBLEMS 329 Table 2. Numerical maximum errors (E N ϵ ) and numerical rate of convergence (p N ) for LSFEM with linear splines: ϵ N = 32 N = 64 N = 28 N = 256 2 63E-3 5E-3.36E-4.967E-5.46 3.3 2.2 2 2.37E-3 5E-3.36E-4.967E-5 2 3.3 2.2 2 3.737E-3 3E-3.473E-4 E-4.39 3.22.49 2 4 5E-2.423E-3.7E-3 69E-4.36.37 2.74 2 5.495E-2.99E-3 25E-3.56E-4.68.4 3.6 2 6 24E-2.43E-2.332E-3.963E-4.75.46 3.96 2 7.98E-2.49E-2.332E-3.963E-4.88.5 3.96 2 8.6E-.57E-2.332E-3.963E-4.9.55 3.96 2 9.E- 2E-2.332E-3.963E-4.92.59 3.96 2 E- 5E-2.332E-3.963E-4.92.6.24 2.3E- 7E-2.332E-3.963E-4.92.6.24 2 2.4E- 7E-2.332E-3.963E-4.92.62.24 2 3.4E- 8E-2.332E-3.963E-4.92.62.24 2 4.4E- 8E-2.332E-3.963E-4.92.62.24 2 5.4E- 8E-2.332E-3.963E-4.92.62.24 E N,δt.4E- 8E-2.332E-3.963E-4 p N.92.62.24 efficient in capturing very sharp boundary layers. In plots 2(c-d), we have used the space-time domain to present the solution profile for ϵ = 2 6 and 2 2 at all time levels. Example 6.2: We consider the nonlinear SPPDE with initial condition u t (x, t) ϵu xx (x, t) + u x (x, t) = e u on Ω = [, ] 2, u(x, ) = for x [, ],

33 V. SANGWAN AND B. KUMAR Table 3. Numerical maximum errors (E N ϵ ) and numerical rate of convergence (p N ) for 3TGFEM with linear splines: ϵ N = 32 N = 64 N = 28 N = 256 2 25E-4 5E-4.538E-5.32E-5.49.38 2 2.39E-3 E-4 2E-4.55E-5 5.8.35.64 2 3.5E-2 86E-3.75E-4.79E-4.39 5.22 -.63 2 4.433E-2.7E-2 67E-3 68E-4.4.39 3.67 2 5 8E-.42E-2.4E-2 59E-3.39.4.39 2 6.33E-.933E-2 86E-2 57E-3.8.8.2 2 7.36E-.3E-.367E-2.5E-2.6.2.6 2 8.39E- 2E-.46E-2.3E-2.6..4 2 9.45E- 6E-.426E-2.37E-2.6.9.3 2.42E- 9E-.435E-2.4E-2.6.8.3 2.46E-.3E-.44E-2.42E-2.6.8.3 2 2.48E-.3E-.442E-2.43E-2.6.8.3 2 3.48E-.3E-.443E-2.44E-2.6.8.3 2 4.49E-.3E-.444E-2.44E-2.6.8.3 2 5.49E-.3E-.444E-2.44E-2.6.8.3 E N,δt.49E-.3E-.444E-2.44E-2 p N.6.8.3 and boundary conditions as u(, t) = for t [, ], u(, t) = for t [, ]. Exact solution is not known in this case. Since we are using Shishkin based logarithmic mesh and for each N, the number of mesh points, the width of the boundary layer region changes, it becomes very difficult to use the double mesh principle for finding the absolute maximum error in the L -norm, though we can calculate at very high computational cost. Therefore, in this case, we only plot the solution profiles for all the three schemes. Again, in this case, since the solution profiles produces same plots, we will plot the solution only for the 3TGFE scheme with exponentially fitted splines, on the validated grid. Only for grid validation test, we plot the solutions for all the three schemes.

3-TGFEM VS LSFEM FOR CONVECTION-DIFFUSION PROBLEMS 33 (a) ε = 2 8, 3TGM - EXP (a) (b) ε = 2, LS - LINEAR (c) ε = 2 2, 3TGM - LINEAR.9.9.9 y ( x, t =.5 ).7.5.4 32 64 28 256 52 y ( x, t =.5 ).7.5.4 32 64 28 256 52 y ( x, t =.5 ).7.5.4 32 64 28 256 52.3.3.3... 5.5.75 5.5.75 5.5.75 (d) N = 64, 3TGM - EXP (e) N = 28, LS - LINEAR (f) N = 28, 3TGM - LINEAR y ( x, t =.9 ).9.7.5.4.3 3TG for ε = 2 4 3TG for ε = 2 8 3TG forε = 2 2 exact forε = 2 4 exact forε = 2 8 exact forε = 2 2 y ( x, t =.7 ).9.7.5.4.3 3TG for ε = 2 5 3TG for ε = 2 9 3TG forε = 2 3 exact forε = 2 5 exact forε = 2 9 exact forε = 2 3 y ( x, t =.5 ).9.7.5.4.3 3TG for ε = 2 6 3TG for ε = 2 3TG forε = 2 4 exact forε = 2 6 exact forε = 2 exact forε = 2 4... 5.5.75 5.5.75 5.5.75 Figure In Fig. 3(a-c), solution plots have been shown for all the three schemes over the grids of 32, 64, 28 and 256 number of mesh elements for different values of ϵ. It can be observed that a grid of 28 elements is sufficient enough to capture the boundary layers for all the three schemes. In plot 3(d), the singular perturbation ϵ-effect has been drawn. One can observe that the boundary layer gets very sharp as ϵ. One can also notice that beyond ϵ, the degree of variation in the sharpness of the boundary layer gets invisible. Fig. 3 shows that over the proposed grids, the schemes are capable enough to handle very sharp boundary layers. Fig. 4(a-d) presents the solution profiles with respect to different time levels. These plots show the evolution of boundary layers for a fixed ϵ as time increases. In plots 4(c-d), we have used the space-time domain as x-y domain and the computed solution is drawn in the third dimension(z-dimension). Plots 3(b,d) explores the power of the proposed schemes in handling very sharp boundary layers. Plots 4(a-b) shows the solution graphs at fixed alternate time levels, whereas plots 4(cd) provides the continuous solution profiles for t = to t =. These plots also provides a platform for understanding the boundary layer phenomenon more clearly. 7. Conclusion: In the present study, the three schemes namely (a) 3TGFEM with exponentially fitted splines, (b) explicit LSFEM with linear splines and (c) 3TGFEM with linear splines were proposed for solving the SPPDE. Since in the boundary layer region,

332 V. SANGWAN AND B. KUMAR (a).9.7 Ν = 64, ε = 2 5, 3TGM - EXP TG3 at t =. TG3 at t =.3 TG3 at t =.5 TG3 at t =.7 TG3 at t =.9 (b).9.7 Ν = 64, ε = 2, 3TGM - EXP TG3 at t =. TG3 at t =.3 TG3 at t =.5 TG3 at t =.7 TG3 at t =.9 u ( x, t ).5.4 u ( x, t ).5.4.3.3. 5.5.75. 5.5.75 (c) Ν = 28, ε = 2 6, 3TGM - EXP Z (d) Ν = 28, ε = 2 2, 3TGM - EXP Z X Y X Y computed solution.4 computed solution.4.4 time.4 space.4 time.4 space Figure 2 a very fine mesh is expected to capture the boundary layer more sharply, Shishkin based logarithmic mesh has been used for smooth condensing of mesh points in the layer region. In 3TGFEM, time discretization is performed prior to the spatial discretization in order to retain the accuracy gained by using the higher order temporal discretization. On this graded mesh, 3TGFE scheme with exponentially fitted splines is shown to be conditionally ϵ-uniform convergent of first order in space and third order in time. Numerical experiments has been carried out for all the three schemes on both linear and nonlinear SPPDE. Absolute maximum norm errors, numerical rate of convergence and numerical ϵ-uniform rate of convergence are presented in Tables(-3). One can observe from these tables, that the 3TGFEM with exponentially fitted splines is more accurate as compared to the other two schemes. Through the depiction of the sharp boundary layers, it has been shown that the proposed schemes are very efficient in capturing these boundary layers. From these numerical results and solution plots, it is very much clear that the proposed schemes provides a very powerful tool for solving the SPPDE for very small values of the singular perturbation parameter ϵ. 8. Acknowledgement The first author is thankful to the Council of Scientific and Industrial Research (CSIR), India for providing the financial support during the research work. References [] Kevorkian, J. and Cole, J., Perturbation methods in Applied mathematics, Springer-Verlag, 98.

3-TGFEM VS LSFEM FOR CONVECTION-DIFFUSION PROBLEMS 333 (a) ε = 2, 3TGM - EXP (a) (b) ε = 2 9, LS - LINEAR y ( x, t = ).9 32 64 28 256 y ( x, t = ).9 32 64 28 256.7.7 5.5.75 5.5.75 (c) ε = 2 7, 3TGM - LINEAR (d) N = 28, 3TGM - EXP y ( x, t = ).9 32 64 28 256 y ( x, t =.9 ).9.7 3TG for ε = 2 3 3TG for ε = 2 5 3TG for ε = 2 7 3TG for ε = 2 9 3TG for ε = 2 3TG for ε = 2 3.7 5.5.75 5.5.75 Figure 3 [2] Roos, H. G., Stynes, M., Tobiska, L., Numerical methods for singularly perturbed differential equations, Convection-Diffusion and Flow problems, Springer, Berlin, 996. [3] Donea, J. and Huerta, A., Finite element methods for flow problems, Wiley, England, 23. [4] Donea, J., A Taylor-Galerkin method for convective transport problems, Int. J. Numer. Methods Engrg. 2 (984), -2. [5] Lax, P. D. and Wendroff, B., Systems of convervation laws, Comm. Pure Appl. Math., 3 (96), 27. [6] Lax, P. D. and Wendroff, B., On the stability of difference schemes, Comm. Pure Appl. Math., 5 (962), 363. [7] Lax, P. D. and Wendroff, B., Difference schemes for hyperbolic equations with high order of accuracy, Comm. Pure Appl. Math. 7 (964), 38. [8] Jiang, C.B. and Kawahara, M., The analysis of unsteady incompressible flows by a three-step finite element method, Int. J. Numer. Methods in Fluids, 6 (993), 793-8. [9] Jiang, Bo-nan, The least-squares finite element method, Theory and Applications in Computational Fluid Dynamics and Electromagnetics, Springer-verlag, Berlin, 998. [] Evrenosoglu, M. and Somali, S., Least-squares methods for solving singularly perturbed twopoint boundary value problems using Bezier control points, Applied Mathematics Letters, 2 (28), 29-32. [] Jiang, Bo-nan, The least-squares finite element method for incompressible Navier-Stokes problems, Internat. J. Numer. Methods Fluids, 4(7) (992), 843-859. [2] Donea, J., Roig, B. and Huerta, A., High-order accuarate time-stepping schemes for convection-diffusion problems, Comput. Methods Appl. Mech. Engrg. 82 (2), 249-275. [3] Miller, J.J.H., O Riordan, E. and Shishkin, G.I., Fitted Numerical methods for singular perturbation problems, World Scientific, Singapore, 996. [4] Farrel, P.A., Hegarty, A.F., Miller, J.J.H., O Riordan, E. and Shishkin, G.I., Robust Computational Techniques for Boundary Layers, Chapman and Hall, CRC Press, Boca Raton, USA, 2.

334 V. SANGWAN AND B. KUMAR (a) u ( x, t ).9 Ν = 28, ε = 2 5, 3TGM - EXP TG3 at t =. TG3 at t =.3 TG3 at t =.5 TG3 at t =.7 TG3 at t =.9 (b) u ( x, t ).9.7 Ν = 28, ε = 2 3, 3TGM - EXP TG3 at t = TG3 at t =.4 TG3 at t = TG3 at t = TG3 at t =..7 5.5.75.5 5.5.75 (c) Ν = 28, ε = 2 4, 3TGM - EXP Z (d) Ν = 28, ε = 2 2, 3TGM - EXP Z X Y X Y computed solution.9.7.4 time.4 space computed solution.4 time.4 space Figure 4 [5] Sangwan, V., Rathish Kumar, B.V., Murthy, S.V.S.S.N.V.G.K. and Nigam, M., Three Step Taylor Galerkin Method for Singularly Perturbed Generalized Hodgkin-Huxley Equation, International Journal of Modelling, Simulation and Scientific Computing, (2) (2), 257-276. [6] Bobisud, L., Second-order linear parabolic equations with small parameter, Arch. Rational Mech. Anal. 27 (967), 385-397. [7] Kellogg, R.B. and Tsan, A., Analysis of some difference approximations for a singular perturbation problem without turning points, Math. Comp. 32 (978), 25-39. [8] Stynes, M. and O Riordan, E., L and L uniform convergence of a difference scheme for a semilinear singular perturbation problem, Numer. Math. 5 (987), 59-53. [9] Stynes, M. and O Riordan, E., Uniformly convergent difference schemes for singularly perturbed parabolic diffusion-convection problems without turning points, Numer. Math. 55 (989), 52-544. [2] Amiraliyev, Gabil M., Erkan Cimen, Numerical method for a singularly perturbed convectiondiffusion problem with delay, Applied Mathematics and Computation, in press, 2. School of Mathematics and Computer Applications,, Thapar University, Patiala 474, India E-mail: sangwan.vivek@gmail.com Department of Mathematics and Statistics, Indian Institute of Technology,, Kanpur 286, India E-mail: bvrk@iitk.ac.in