Manoj Kumar 1, Pratibha Joshi 2. (Received 31 August 2011, accepted 26 December 2011)

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1 ISSN (print), (online) International Journal of Nonlinear Science Vol.14(2012) No.1,pp A Mathematical Model and Numerical Solution of a One Dimensional Steady State Heat Conduction Problem by Using High Order Immersed Interface Method on Non- Uniform Mesh Manoj Kumar 1, Pratibha Joshi 2 1 Department of Mathematics, Motilal Nehru National Institute of Technology, Allahabad, India 2 Department of Mathematics, Graphic Era University, Dehradun, India (Received 31 August 2011, accepted 26 December 2011) Abstract: Elliptic interface problems arise in various areas of science and engineering. The steady state heat conduction in layered bodies is one of the largest areas of application of elliptic interface problems. In this paper, a mathematical model and solution of a one dimensional elliptic interface problem which represents a steady state heat conduction problem in composite medium have been discussed by using high order immersed interface method on non uniform mesh. Numerical results show good agreement with exact solution. Keywords: Elliptic interface problems; Steady state heat conduction; Immersed interface methods; Adaptive mesh refinement. 1 Introduction Elliptic interface problems are basically elliptic boundary value problems with discontinuous coefficients or solution across some interface. Standard finite difference methods may not give accurate results for these problems especially near the interface because of the discontinuity. From past few years many mathematicians and researchers are working on developing methods for interface problems. Initially the popular methods were Peskin s immersed boundary method, the smoothing method, the method of harmonic averaging. In 1994, Z. Li and LeVeque [15] developed original immersed interface method in which they incorporated the jump conditions into the finite difference method. The method was second order accurate. The method adds some additional nodes to the numerical stencil which may lead to a nonsymmetric matrix which can not be solved by standard numerical solvers. Since then many researchers have proposed variety of methods [1 14, 16]. Some of them are developed with the modifications in the standard methods so that they can deal with the discontinuities and the singularities. In [9], a decomposed immersed interface method is introduced where a correction term is introduced to standard difference stencil on the right hand side only so that the resulting linear system remains symmetric and diagonally dominant and can be solved by standard solvers. The correction term is decomposed in both Cartesian directions. Another method EJIIM (Explicit Jump Immersed Interface Method)is presented in [10], where high order jumps at the intersection of the interface and coordinate system are introduced as auxiliary unknowns.the interpolation equation is derived for the high order jumps by taking one sided local approximation and the information about jumps. In [10], the grid generation by correcting the finite difference scheme near the interface is avoided. EJIIM was actually motivated by fast iterative IIM introduced by Z. Li in which elliptic equation is preconditioned before using the immersed interface method. There is another efficient approach Matched interface and boundary method [11], in which the solution on each side of the interface is smoothly extended along the interface by some fictitious points. The fictitious values on the fictitious are evaluated from enforcing the jump conditions at the exact position of the interface. Many other approaches have also been developed like finite element method, integral equation method, discontinuous Galerkin approach etc. Elliptic Interface problems have various physical and engineering applications, such as heat conduction in layered medium, alloy solidification, crystal growth, cell and bubble formation, biochemical processing. Corresponding author. address: pratibha.joshi@gmail.com Copyright c World Academic Press, World Academic Union IJNS /633

2 12 International Journal of Nonlinear Science, Vol.14(2012), No.1, pp Heat conduction in composite medium is a wide area of application of elliptic interface problems. Mathematical modelling of heat conduction in composite walls, fins, layered conductor leads to an elliptic interface problem with discontinuous coefficient. In many engineering problems knowledge of temperature distribution and heat flux in materials composed of different layers with different thermal conductivities e.g. construction of furnaces, conductors, thermal protection applications, laminated and fibre reinforced materials, cooling problems is needed. In addition, steady state heat conduction is very helpful for the study of the structural integrity of layered devices in electronic applications. There have been many studies previously on developing techniques for analytical and numerical solution of problems related to steady state heat conduction. [1 3, 12]. In this paper we have first discussed the Zhong s high order IIM [5] and then generalised the method on non uniform mesh and derived the difference formula. Then we applied a one dimensional adaptive grid generaion algorithm to increase the accuracy. Finally we have solved some steady state heat conduction problems in one dimension. The paper has been divided in five sections. In first section we have discussed the mathematical model of a steady state heat conduction problem.in second section the high order IIM has been described. In third section we have obtained the discretization on non uniform mesh. In fourth section we have discussed the adaptive grid generation algorithm which we are going to apply on the scheme. In the last section we have numerically solved some one dimensional steady state heat conduction problems and illustrated the computational results. 2 Mathematical model of steady state heat conduction in one dimensional layered medium In this section, we will discuss the mathematical interpretation of steady state heat conduction equation and derive the heat conduction equation. Due to the existence of temperature gradient within a body, heat energy will flow from the region of high temperature to the region of low temperature. This phenomenon is known as conduction heat transfer, and is described by Fourier s Law q = k T. This equation determines the heat flux vector q for a given temperature profile T and thermal conductivity k. The minus sign ensures that heat flows down the temperature gradient. The heat equation follows from the conservation of energy for a small element within the body, Heat conducted in + Heat generated within = Heat conducted out + Change in energy stored within We can combine the heats conducted in and out into one net heat conducted out term to give, Net heat conducted out = Heat generated within - Change in energy stored within Mathematically, this equation is expressed as, q = q gen de dt where q gen is the power generated per unit volume and e is the internal energy. The change in internal energy e is related to the body s ability to store heat by raising its temperature, given by, (1) de dt dt = ρc dt Here c is the specific heat and ρ is the density. Also, the thermal diffusivity α is related to the thermal conductivity k,the specific heatc and the density ρ by, α = k ρc Now substituting for q using Fourier s Law of heat conduction from above to arrive at the Heat Equation, IJNS for contribution: editor@nonlinearscience.org.uk

3 M. Kumar, P. Joshi: A Mathematical Model and Numerical Solution of a One Dimensional Steady State Heat Conduction 13 In steady state, the heat equation takes the form ( k T ) = q gen ρc dt dt k 2 T + ρc dt = q gen 2 T 1 α dt dt dt = 1 k q gen ( k T ) = q gen Hence a one dimensional steady state heat conduction problem leads to an elliptic boundary value problem of the following form: (β(x)u (x)) = f(x) xϵω (2) where u(x) is the temperature distribution, β(x) is the heat conductivity and f(x) is the source. The domain is a layered medium i.e. it consists of two layers, say Ω 1 and Ω 2 with different heat conductivities, β 1 (x) and β 2 (x). Suppose their common boundary is at x = a. The above problem can be modeled into an elliptic interface problem with discontinuous coefficient β(x) across the interface x = a. Figure 1: One dimensional layered medium. { β1 (x) x Ω β(x) = 1 β 2 (x) x Ω 2 The domain is Ω = Ω 1 Ω 1. Suppose the temperature and flux have jump discontinuity across the interface which are defined as where [u] x=a = u + u [βu x ] x=a = β + u x + β u x u = lim x a u(x) and u+ = lim x a +u(x) β = lim β(x) and β + = lim β(x) x a x a + If the jumps defined above are equal to zero, the contact between the layers is called ideal. 3 High order immersed interface method to solve elliptic interface problem Xiaolin Zhong presented a high order immersed interface method for the simulation of multi phase flow [5]. The benefits of the method are the following: I. We can derive finite difference scheme of arbitrary order with the same methodology. II. It needs only two physical jump conditions of the variables and their first derivatives to achieve second or higher order accuracy. III. The method is applicable to variety of interface problems since the finite difference formulas can be expressed in explicit form. IJNS homepage:

4 14 International Journal of Nonlinear Science, Vol.14(2012), No.1, pp In the above scheme, standard finite difference formula has been used at regular points whereas at irregular points, special finite difference formulae have been derived by using Taylor expansion and matched polynomial interpolation. The discretization has been done on a uniform mesh. Suppose we have a one dimensional second order elliptic boundary value problem with following jump conditions [αu] = α + u + α u = A (3) [βu x ] = β + u x + β u x = B (4) The interface is at x Γ where x i and x i+1 are the irregular points i.e. x i < x Γ < x i+1. In [5], to derive standard finite Figure 2: Stencil near the interface. difference formulae by using matched polynomial interpolation, two matched polynomials are used to interpolate both sides of the interface. Due to the discontinuity of the solution across the interface single interpolation polynomial can not be used. For discretization, n points on left side of the interface and m points on the right side have been taken which will lead to a n + m points grid stencil. The polynomial interpolating left hand side of the interface can be written as where a n is an undetermined coefficient and P (x) = l k (x)u i+k + a n R(x) (5) n+1 k=0 R(x) = (x x i+k ) (6) n+1 l k (x) is the Lagrange polynomial interpolating n points on left side of the interface n+1 l=0,l k l k (x) = (x x i+l) n+1 l=0,l k (x i+k x i+l ) k=0 The polynomial for the right hand side of the interface can be written in similar manner as m P + (x) = h k (x)u i+k + b m Q(x) (8) where b m is an undetermined coefficient and k=1 Q(x) = m (x x i+k ) (9) k=1 (7) IJNS for contribution: editor@nonlinearscience.org.uk

5 M. Kumar, P. Joshi: A Mathematical Model and Numerical Solution of a One Dimensional Steady State Heat Conduction 15 h k (x) is the Lagrange polynomial interpolating m points on the right hand side m l=1,l k h k (x) = (x x i+l) m l=1,l k (x i+k x i+l ) (10) The undetermined coefficients a n and b m are decided by the jump conditions. Using the values of (5) and (8) in jump conditions (3) and (4) we have and ( m ) α + h k (x Γ )u i+k + b m Q(x Γ ) k=1 ( m ) β + h k(x)u i+k + b m Q (x) k=1 Solving equations (11) and (12) we get a n = m k= n+1 ( n+1 α k=0 ( n+1 β k=0 γ k u i+k + ξ A A + ξ B B l k (x Γ )u i+k + a n R(x Γ ) l k(x)u i+k + a n R (x) ) ) = A (11) = B (12) where and { 1 J γ k = (α β + Q (x Γ )l k (x Γ )) α + β Q(x Γ )l k (x Γ)) (k = n + 1,..., 0) α + β + J ( Q (x Γ )h k (x Γ ) + Q(x Γ )h k (x Γ)) (k = 1,..., m) ξ A = 1 J β+ Q (x Γ ) ξ B = 1 J α+ Q(x Γ )) J = α + β R (x Γ )Q(x Γ ) α β + R(x Γ )Q (x Γ ) Similarly coefficientb m can be evaluated. Value of b m is not determined here because it is not needed for evaluating difference scheme at irregular point i. Hence the r th derivative at irregular point i can be approximated as ( d r ) ( y d r P ) (x) dx r i dx r (13) x=x i Once we have determined the undetermined coefficients we get the full expression for the polynomials. We can derive expression for any order of derivative from the equation (13). Hence differentiating the polynomial r times we get ( d r ) y dx r i m k= n+1 d k u i+k + d A A + d B B (14) where d k = { l (r) k x(i) + γ kr (r) x(i) (k = n + 1,..., 0) γ k R (r) x(i) (k = 1,..., m) d A = ξ A R(r) x(i) d B = ξ B R(r) x(i) This is the finite difference formula for the irregular point i on the left hand side of the interface. We can get the difference formula at irregular point i + 1 by a coordinate transformation as described in [5]. At regular points we can use the standard central difference formula. u xx = u i 1 2u i + u i+1 h 2 IJNS homepage:

6 16 International Journal of Nonlinear Science, Vol.14(2012), No.1, pp High order immersed interface method on non uniform mesh In the scheme [5] the mesh is taken uniform. For n = m = 2 the difference scheme (14) gives the approximation at irregular point i as Here d k s are defined as where du = d 1u i 1 + d 0 u 0 + d 1 u i+1 + d 2 u i+2 + d A A + hd B B + u i u i 1 dx 2h h ( d 2 ) u dx 2 = d 1u i 1 + d 0 u 0 + d 1 u i+1 + d 2 u i+2 + d A A + hd B B i h 2 (15) d 1 = 1 ( ( cα 3σ 2σ 2 ) ( c β 2 + 3σ σ 2 )) D d 0 = 1 ( ( cα 3 σ + 2σ 2 ) ( c β 2 3σ + σ 2 )) D d 1 = 1 ( 4 4σ + σ 2 ) D d 2 = 1 ( 1 + 2σ σ 2 ) D d A = 1 α + ( 3 + 2σ) D d B = 1 ( 2 3σ + σ 2 ) β + D D = 1 2 c α = α α + c β = β β + ( cβ ( 2 + σ 5σ 2 + 2σ 3) c α ( 3σ σ 2 + 2σ 3)) Here h is the uniform mesh length. Now we propose the discretization on non uniform mesh. Suppose we have a grid {x i }, i = 1, 2,... with the grid lengths {h i }, i = 1, 2,... where h i = x i+1 x i and interface is located at x Γ = x i + σh. Then we can approximate the second order derivative of unknown as Figure 3: Four point stencil taking two points on both side of the interface. ( d 2 ) u dx 2 = t 1 u i 1 + t 0 u 0 + t 1 u i+1 + t 2 u i+2 + t A A + t B B i IJNS for contribution: editor@nonlinearscience.org.uk

7 M. Kumar, P. Joshi: A Mathematical Model and Numerical Solution of a One Dimensional Steady State Heat Conduction 17 The t k s are defined as T = 1 2 (c β (2σh i + h i 1 ) ((1 σ)h i ((1 σ)h i + h i+1 )) + c α (2(1 σ)h i + h i+1 ) (σh i (σh i + h i 1 ))) σh c α (2(1 σ)h i + h i+1 ) i 1 h i 1 + c β (((1 σ)h i )((1 σ)h i + h i+1 )) h i 1 t 1 = T c α ( 2(1 σ)h i h i+1 ) hi 1+σh i 1 h i 1 c β (((1 σ)h i )((1 σ)h i + h i+1 )) h i 1 t 0 = T (2(1 σ)h i + h i+1 ) ((1 σ)hi )+h i+1 1 h i+1 (((1 σ)h i )((1 σ)h i + h i+1 )) h i 1 t 1 = (16) T ( 2(1 σ)h i + h i+1 ) (1 σ)hi 1 h i+1 + (((1 σ)h i )((1 σ)h i + h i+1 )) h i 1 t 2 = T t A = β+ ( 2(1 σ)h i h i+1 ) T t B = α+ ((1 σ)h i ((1 σ)h i + h i+1 )) T We can find the approximation for the irregular point i + 1 by a coordinate transformation x = x As a result we get the finite difference formula as ( d 2 ) u dx 2 = t 2 u i 1 + t 1 u 0 + t 0 u i+1 + t 1 u i+2 t A A + t B B (17) i where t k s can be determined by puttingσ = 1 σ and interchanging places of h i+1 and h i 1 in equations (16). 5 Adaptive grid generation in the immersed interface method Adaptive grid generation is a popular and efficient method to increase accuracy of difference schemes. Many researchers have done significant work in generating and applying adaptive grids in the numerical solution of differential equations [7, 8]. It is basically special type of grid generation where we add some more grid points in the interval in which the solution has large variations such as peaks or boundary layers etc. In interface problems generally the interval having largest variation in the solution is the interval containing the interface. We have applied this concept in the difference scheme [5]. Since the intervals are not of uniform length after the adaptive grid generation; we have determined the scheme on non uniform mesh. There are various methods to generate adaptive grids. The common property of most of these methods is that they divide the solution domain into subintervals such that some positive weight function has approximately same value over each subinterval. The choice of the weight function depends on the first derivative or the second derivative or the truncation error. The algorithm we have applied in the difference scheme above is presented in [6]. The algorithm has the following steps: I. Generate a uniform mesh in the given solution domain. II. Compute the error in each subinterval. III. Determine the subinterval of maximum error. IV. Include the middle point into that subinterval. V. Repeat the procedure until the stopping criteria is satisfied. We have applied the algorithm on the interval containing the interface. Since the immersed interface method developed in [5] is built on uniform grids, it is not possible to apply the adaptive grid generation in the same scheme. Therefore, we have determined the difference scheme on non uniform mesh. At regular point we have used the standard finite difference formula on non uniform mesh. u xx = 2[h i 1u(x i+1 ) (h i + h i 1 )u(x i ) + h i u(x i 1 )] h i h i 1 (h i + h i 1 ) IJNS homepage:

8 18 International Journal of Nonlinear Science, Vol.14(2012), No.1, pp We will prove the efficiency of the proposed scheme in next section by solving one dimensional heat conduction problems. The results are very satisfactory. 6 Numerical results To illustrate our proposed scheme we consider two one dimensional steady state heat conduction problems. Suppose we have a very thin rod of unit length constructed with two materials having different conductivities. The contact of the two layers is at x = α.i.e. { k1 x < α k = x > α where k 1 and k 2 are constants. k 2 Problem 1 In this problem the temperature distribution is governed by the following elliptic problem (ku x ) x = 12x 2, 0 < x < 1 where the temperature at end points of the rod is given by u(0) = 0, u(1) = 1 ( ) α. k 2 k 1 k 2 The jump conditions for the temperature and the flux along the interface are The exact solution of the problem is u(x) = [u] x=α = 0, [ku x ] x=α = 0 x 4 k 1 x 4 k 2 + ( 1 k 1 1 k 2 )α 4 x < α x > α Now we apply the new immersed interface method described in section 4 and solve the problem by taking different values of.we have developed a computer program of the proposed method applied on the above problem in MATLAB 7. The algorithm for adaptive grid generation discussed in section 5 terminates when some stopping criterion is satisfied which can be either required level of accuracy or number of subintervals etc. In our MATLAB program, number of subintervals required has been chosen as the criterion to terminate. The order of the scheme can be evaluated by en log p = e 2n log(2) (18) where e n is the maximum norm defined by e n = max u(x i ) U(x i ) (19) i where u(x) is the approximate solution and U(x) is the exact solution of the problem. The error ratio can be evaluated by e n e 2n. (20) The grid refinement analysis is displayed in Table I. We have taken three different sets of constants and evaluated the maximum absolute error in each case. We have taken two meshes for N = 10 and N = 20. In the proposed method the mesh is unequally distributed whereas in IIM [5] the mesh has equal subintervals. The order of convergence and the error ratio can easily be evaluated by the formulas. In general the order of proposed method is O(h 2 ). The resulting IJNS for contribution: editor@nonlinearscience.org.uk

9 M. Kumar, P. Joshi: A Mathematical Model and Numerical Solution of a One Dimensional Steady State Heat Conduction 19 Figure 4: Comparison of numerical solution by proposed IIM and exact solution of Problem-1 for k 1 = 1, k 2 = 100, α = 2/3 and N = 20 unequal subintervals. Figure 5: Comparison of numerical solution by proposed IIM and exact solution of Problem-1 for k 1 = 1, k 2 = 1000, α = 2/3 and N = 20 unequal subintervals. Table 1: Maximum Absolute Error. Constants e n (Proposed IIM) N = 10 N = 20 k 1 = 1, k 2 = 100, α = 2/ k 1 = 1, k 2 = 500, α = 2/ k 1 = 1, k 2 = 1000, α = 2/ IJNS homepage:

10 20 International Journal of Nonlinear Science, Vol.14(2012), No.1, pp system of equations comes nonsymmetric in both cases because of the change of difference scheme at irregular points. However, the facts that the method is very flexible to apply in different interface problems and one can easily construct high order schemes from the given schemes (in Section 2 and 3) make the methods overweight the lose of symmetry in the resultant matrix. We have used the operator in MATLAB for solving the nonsymmetric coefficient matrix because it is very popular among the people working in MATLAB because of its efficiency. Recently many multigrid solvers and iterative methods are being used for nonsymmetric sparse system of equations which gives fast and accurate solution e.g. BiConjugate Gradient Stabilized (BiCGSTAB), Generalized Minimum Residual Method - an orthogonalization and least squares method (GMRES) etc which can also be used here instead of the direct solver (\ in MATLAB). The computational errors of the above methods depend on the location of the interface i.e. the value of σ.the value of σ changes with changes in number of subintervals. Problem 2 In this problem the temperature distribution is governed by the following equation with boundary conditions and thermal conductivities of the two materials are x (k x u(x)) = 0, x (0, 1) u(0) = 0, u(1) = 1 k = { 1 x < α 2 x > α The exact solution of the problem is 2x x < α u(x) = α + x 1 q x > α This problem is solved on two non uniform meshes with 21 and 41 intervals. Computational results are given in Table II. We have considered three cases of the problem with three different positions of the interface. In each situation the proposed method works well. Figure 6: Comparison of numerical solution by proposed IIM and exact solution of Problem-2 for N = 41 unequal subintervals at α = 6/7. Table 2: Maximum Absolute Error. Position of the interface e n (Proposed IIM) N = 21 N = 41 α = 3/ α = 6/ α = 7/ The computational results show good agreement with the exact solution for both the problems. Mostly the proposed method gives better result than the high order IIM [5]. The adaptive grid is constructed as to give the required accuracy. But one has to take care as sometimes the convergence could be lost while constructing it. IJNS for contribution: editor@nonlinearscience.org.uk

11 M. Kumar, P. Joshi: A Mathematical Model and Numerical Solution of a One Dimensional Steady State Heat Conduction 21 Figure 7: Comparison of numerical solution by proposed IIM and exact solution of Problem-2 for N = 41 unequal subintervals at α = 3/4. 7 Conclusion We have discussed the mathematical model of one dimensional steady state heat conduction problems and solved them by using high order IIM on non uniform grid applying the one dimensional adaptive grid generation algorithm described in [6]. The proposed method is well suited for the above problems.the proposed non uniform discretization can be used with many non uniform grids and adaptive grids which produces more accurate results mostly. In [5], Xiaolin Zhong was motivated to derive the high order IIM because of the need of an efficient method to solve problem in two phase flow. However the method is applicable to some steady state heat conduction problems too. The advantage of using these methods in steady state heat conduction problem is that they need only two jump condition and they are easy to apply on different types of problems. We can easily generate difference scheme of arbitrary order by the procedure described in 3. References [1] L.G. Stanley. A Senstivity Equation method for Molding Processes. Proceedings of the 2000 IEEE International Conference on Control Applications Anchorage, Alaska, USA,September [2] Z. M. Seyidmamedov and E. Ozbilge. A mathematical model and numerical solution of interface problems for steady state heat conduction. Mathematical Problems in Engineering., 2006(2006):1 18. [3] Okey Oseloka Onyejekwe. Heat conduction in composite media: a boundary integral approach. Computers and Chemical Engineering., 26(2002): [4] Z. Li. The Immersed Interface Method - A Numerical Approach for Partial Differential Equations with Interfaces. Ph.D. thesis, University of Washington [5] Xiaolin Zhong. A new high-order immersed interface method for solving elliptic equations with imbedded interface of discontinuity. Journal of Computational Physics., 225(2007): [6] M. M. Ahmed Khodier and Y. Adel Hassan. One dimensional adaptive grid generation. Internat. J. Math. & Math. Sci., 20(1997)(3): [7] J. F. Thompson. A Survey of Dynamically Adaptive Grid in the Numerical Solution of Partial Differential Equations. Appl. Numer. Math., (1985):3 27. [8] V. E. Denny and It. B. Landis. A New Method for Solving Two Point Boundary Value Problems Using Optimal Node Distribution. J. of comp. Phys., 9(1972): [9] Berthelsen,Andreas Petter. A decomposed immersed interface method for variable coefficient elliptic equations with non-smooth and discontinuous solutions. Journal of Computational Physics, 197(2004): IJNS homepage:

12 22 International Journal of Nonlinear Science, Vol.14(2012), No.1, pp [10] A. Wiegmann and K. Bube. The explicit-jump immersed interface method: finite difference methods for PDEs with piecewise smooth solutions. SIAM J. Numer. Anal., 37(2000): [11] Y. C. Zhou and G. W. Wei. On the fictitious -domain and interpolation formulations of the matched interface and boundary method. Journal of Computational Physics., 219(2006): [12] S. Naser Al-Huniti and M. A. Al-Nimr. Steady-State Thermoelastic Behavior of a Two-Anisotropic Layer Thick Plate Strip. International Journal for Computational Methods in Engineering Science and Mechanics., 7(2006): [13] Sumir Chandra and Manish Parashar. Addressing spatiotemporal and computational heterogeneity in structured adaptive mesh refinement. Computing and Visualization in Science., 9(2006): [14] R. Kornhuber, R. Krause, O. Sander, P. Deuflhard and S. Ertel. A monotone multigrid solver for two body contact problems in biomechanics. Computing and Visualization in Science., 11(2008):3 15. [15] R. J. LeVeque and Z. Li. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal., 31(1994): [16] Manoj Kumar and Pratibha Joshi. Some Numerical Techniques for Solving Elliptic Interface Problems. Numerical Methods for Partial Differential Equations., 28(2012)(1): [17] Pratibha Joshi, Manoj Kumar. Mathematical Model and Computer Simulation of Three Dimensional Thin Film Elliptic Interface Problems. Computers and Mathematics with Applications,63(1)(2012): IJNS for contribution: editor@nonlinearscience.org.uk