An Ultra Weak Finite Element Method as an Alternative to a Monte Carlo Method for an Elasto-Plastic Problem with Noise

An efficient method for obtaining numerical solutions of an elasto-plastic oscillator with noise is considered. Since Monte Carlo simulations for the underlying stochastic process are too slow, as an alternative, approximate solutions of the partial differential equation defining the invariant measure of the process are studied. The regularity of the solution of that partial differential equation is not sufficient to employ a “standard” finite element method. To overcome the difficulty, an ultra weak finite element method has been developed and successfully implemented.