The Approximation Solution of a Nonlinear Parabolic Boundary Value Problem Via Galerkin Finite Elements Method with Crank-Nicolson

This paper deals with finding the approximation solution of a nonlinear parabolic boundary value problem (NLPBVP) by using the Galekin finite element method (GFEM) in space and Crank Nicolson (CN) scheme in time, the problem then reduces to solve a Galerkin nonlinear algebraic system(GNLAS). The predictor and the corrector technique (PCT) is applied here to solve the GNLAS, by transforms it to a Galerkin linear algebraic system (GLAS). This GLAS is solved once using the Cholesky method (CHM) as it appears in the matlab package and once again using the Cholesky reduction order technique (CHROT) which we employ it here to save a massive time. The results, for CHROT are given by tables and figures and show the efficiency of this method, from other sides we conclude that the both methods are given the same results, but the CHROT is very fast than the CHM.


Introduction
In the last decades many researchers interested to study the solution of boundary value problems (bvps) in general and the solution of NLPBVP in particular, there are many different methods for solving the NLPBVP, f.g. in 2000, Karlsen and Riserbo used a corrected operator Splitting method [1], Pao in 2001 used the time period solutions [2], in 2006, Alam and etc used the simultaneous space-time adaptive wavelet method [3]. Timothy in 2010 studied the explicit and implicit difference method [4], in 2011 Ghoreishi and Ismail are used the Homotopy Perturbation Method (HPM) [ 5], and many others.
The study of the solution for the parabolic bvp using the finite element method (FEM) back to the beginning of the 17 th century, and are studied from many researchers so as Douglas and Dupont [6], in 1993 Reddy introduced in his book an introduction to the FEM was applied to linear, one and two-dimensional problems of engineering and applied sciences [7]. In 1997-2006 Thomee [8] studied the GFEM with backward Euler method for nonlinear parabolic bvp. According to these studied it was important in this paper to study the approximate solution for NLPBVP using the GFEM method for the space variable and the CN scheme for the time variable. This paper starts with give a description of proposed NLPBVP and its weak form (wf). The approximation solution of the problem is obtained by discretize the wf by using the GFEM for the space variable and the CN Scheme for the time variable, the problem then reduces to solve a GNLAS which transforms it to a LAS which is solved once using the CHM and once again using that we give it the name CHROT to save a massive time, which is explained in a two steps formula. Finally, illustrative examples are given to solve different problems using MATLAB R2013a software, the results are given by tables and by figures and are show the efficiency of this method, and are show that the CHROT is very fast to solve the linear algebraic system than the CHM. Definition 1, [9]: A point * ∈ ⊂ ℝ is said to be fixed point of a given function : → ℝ , if * * .
In this work the inner product and norm in L W will be denoted by • ,• and ‖•‖ , the inner product and norm in Sobolev space V H W will be denoted by • , • and ‖•‖ , the duality bracket between V and its dual V * will be denoted by 〈• ,•〉 and ‖•‖ be the norm in L P . Now, the wf of (1-3) is given by:

4.Discretization of the Continuous Equation:
The wf of (4-5) is discretized by using the GFEM, as follows, let the domain W is a polyhedron. For every integer , let be an admissible regular traingulation of W into closed disimplices [8], be a subdivision of the interval I ̅ into intervals , where , , of equal length ∆ ⁄ , let and ⊂ W be the space of continuous piecewise affine in W. The Discrete state equation (DSEq) of the wf (4-5) is obtained after using the CN formula and is given by where ∈ , , , j=0,1 ...,NT, ∈ ∈ , ∀ 0,1, … , .

The Approximation Solution of the Nonlinear Parabolic Equation:
To find the approximation solution (app.sol) , , … , of (6-7) by using the GFEM, the following procedure can be used: (1) For fixed any , 0 1 , let , 1,2, … , , ℎ ⃗ 0 , be a continuous piecewise affine finite basis of in W , then for any i=1,2,...,N and , ∈ (6-7) can be rewritten as: , , , ∈ , (9) (2) Using the Galerkin method [8], with the basis , , … , of , one has ∑ , , ∑ , and ∑ where , , for each 0,1, … , are unknown constants to be determine. (8) to get the following nonlinear algebraic system and substituting in (9) to get the following linear algebraic system (11) where , System (10-11) has a unique solution [10]. To solve it, the linear algebraic system (11) is solved at first to get , then to solve the nonlinear system (10) the PCT is used here [8], as follows : For each 0 1 we predict at first the value by using the explicit form (just the value of ) in the component of ⃗ in the right hand side (RHS) of (10) , then by substitute , in the component of ⃗ in the RHS of (10), makes system (10) linear, solving it w.r.t to get the corrector solution , this procedure can be repeated (more than one time if we need) by substitute the corrector solution in the RHS of the linear system (10) and solve it again w.r.t , for fixed , to get a new corrector solution). Hence the corrector equation described as follows: where ∶ is the predictor solution at the iteration 1, ∶ is its corresponding corrector solution at the iteration and is the known corrector solution for the previous step j, i.e. (12) can be written as : (13) Theorem 6: The discrete state Equation (6-7) with fixed point and for ∆ sufficiently small has a unique solution , , … , , and the sequence of corrector solutions is convergence in ℝ.
By subtracting (15) from (14), setting in the obtained equation and using Lipschitz condition on with respect to for , once get that Keep in mind that the 2 term in the left hand side (LHS) is positive and then using Cauchy Schwarz (CS) inequality on the RHS of above inequality, once get that , where ∆ , using (13), to get that Since ∆ is sufficiently small and 1, then q is contractive, and by theorem (1) we get , hence the DSEq has a unique solution, also since ∈ ℝ , ∀ then ∈ ℝ , ∀ implies that ∈ , ∀ ∈ ℝ , and by using Theorem (3) with S=ℝ , we get that is converged to a point in ℝ .

Cholesky Reduction Order Technique
This technique in fact is based on an idea which is introduce first in [11] about reducing the diagonal elements of the Galerkin matrix into columns, we formulate it by the following steps hence we called it by the Cholesky reduction order technique (CHROT): First, the obtained matrix is reduced to 1 matrix by transform the lower diagonals ( 1 of matrix to columns, second the reduction of matrix is a new 1 matrix which is computed by using the following formula : ⃗, 1 1 This problem solved using the GFEM for M=9 and NT=20, the results are shown in Table 1. and Figure 1. at time ̂ 0.5 , the table shows the approximate solution , , , the exact solution , , and the absolute error at & . The Mat lap Software is used to solve this problem, it takes 5-hours when we use the CHM, while takes 1-hour and 7minutes when we use the CHROT.  This problem is solved using the GFEM for M=9 and NT=20, the results are shown in Table  2. and Figure 2. at ̂ 0.5 , the table shows the approximate solution , , , the exact solution , , and the absolute error at & . This problem is take 13 hours when we use the CHM to solve the PCT, while takes 3 hours and 27 minute when we use the CHROT.

Conclusion
• The GFEM associated with the PCT is suitable, efficient and very fast to solve the nonlinear parabolic boundary value problems.
• The CHROT is very fast than the CHM with same results and this is important when we have problems gives very large algebraic systems which take a long time in the classical CHM.
• The value of ̂ is chose arbitral in the interval I , same results with same accuracy will obtained if we can take any other value of ̂ provided this value belong to I.