Simulation of a 4^th order ODE: Illustration of various primal & mixed MLPG methods

Various MLPG methods, with the MLS approximation for the trial function, in the solution of a 4^th order ordinary differential equation are illustrated. Both the primal MLPG methods and the mixed MLPG methods are used. All the possible local weak forms for a 4^th order ordinary differential equation are presented. In the first kind of mixed MLPG methods, both the displacement and its second derivative are interpolated independently through the MLS interpolation scheme. In the second kind of mixed MLPG methods, the displacement, its first derivative, second derivative and third derivative are interpolated independently through the MLS interpolation scheme. The nodal values of the independently interpolated derivatives are expressed in terms of nodal values of the independently interpolated displacements, by simply enforcing the strain-displacement relationships directly by collocation at the nodal points. The mixed MLPG methods avoid the need for a direct evaluation of high order derivatives of the primary variables in the local weak forms, and thus reduce the continuity-requirement on the trial function. Numerical results are presented to illustrate the effectiveness of the primal, as well as two kind of mixed MLPG methods. It is concluded that the mixed MLPG methods are very costeffective.