EFG For Some Evolutionary Variational Inequalities And Its Convergence Analysis

Meshless method is a kind of popular numerical method in recent decades. TheElement-free Galerkin meshless method (EFG) is an important prolongation of meshlessmethod. The convergence analysis for EFG method of some variational inequalities isintroduced in this paper and applied to solve some evolutionary variational inequality (EVI)problems. Numerical results reveal the accuracy and efficiency of the convergence analysistheorems.The main contents in this paper are as follows:(1) The basic principles of the MLS approximation are introduced. Using linear elasticproblem as an example, it demonstrated the procedure of EFG. The numerical examplefor the Poisson problems and heat conduction problem are realized, and the effect ofvarious parameters is discussed.(2) EFG for a kind of time-dependent parabolic variational inequality (PVI) is discussed. Ityields the EFG fully discrete format and its convergence theorem. The convergenceorder for the PVI is obtained, and it shows that the convergence order depends on notonly the number of basis function in MLS but also the time step or spatial step.Through a heat-servo control problem, it verifies the results of the convergencetheorem.(3) EFG for a kind of time-dependent (second order) EVI is discussed. First it obtained thetheorem of existence and uniqueness for the EVI, next it yields the EFG fully discreteformat and its convergence theorem. The convergence order for the EVI is alsoobtained, and it shows that the convergence order not only depends on the number ofbasis function in MLS but also depend on the relationship between the time step andthe spatial step. By solving an example problem, it verifies the results of theconvergence theorem.