Least-squares finite element schemes in the time domain Original Research Article

Least-squares finite element procedure is used to generate recurrence relations for numerical solution of system of ordinary differential equations of the first order. One-step least-squares method due to O.C. Zienkiewicz and R.W. Lewis [Earthquake Engrg. Struct. Dyn. 1 (1973) 407–408] is reviewed. An analysis of stability and other numerical properties of this method is presented, and it is found to be A-stable and second-order accurate. Using quadratic time element with Lagrange interpolation functions, a two-step least-squares method is derived. Analysis of local discretization error, stability and other properties of the two-step method is presented. It is found that the two-step least-squares algorithm is third-order accurate and A(α)-stable (α≈85∘). Comparison of numerical results obtained with the least-squares schemes with those obtained with other well known algorithms shows that the two-step least-squares scheme and three-step backward-difference scheme exhibit almost the same accuracy, whereas the one-step least-squares scheme is more accurate than one-step θ-methods and two-step backward-difference scheme. Further, the least-squares schemes exhibit superior accuracy at large time values for the problems tending towards a steady state.