Approximation Solution of Nonlinear Parabolic Partial Differential Equation via Mixed Galerkin Finite Elements Method with the Crank-Nicolson Scheme
The approximate solution of a nonlinear parabolic boundary value problem with variable coefficients (NLPBVPVC) is found by using mixed Galekin finite element method (GFEM) in space variable with Crank Nicolson (C-N) scheme in time variable. The problem is reduced to solve a Galerkin nonlinear algebraic system (NLAS), which is solved by applying the predictor and the corrector method (PCM), which transforms the NLAS into a Galerkin linear algebraic system (LAS). This LAS is solved once using the Cholesky technique (CHT) as it appears in the MATLAB package and once again using the General Cholesky Reduction Order Technique (GCHROT), the GCHROT is employed here at first time to play an important role for saving a massive time. Illustrative examples are given to solve the NLPBVPVC with the GCHROT, the results are given by tables and figures which show from a side efficiency of this technique, and from another side show that the two methods GCHROT and CHM are given the same results, but the suggesting first technique is very fast than the second one.