Uniformly Convergent 3-tgfem Vs Lsfem for Singularly Perturbed Convection-diffusion Problems on a Shishkin Based Logarithmic Mesh

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.