Research On An LDG Method For Time-Dependent Fourth-Order Problems With Neumann Boundary Conditions

In this work,we analyze a local discontinuous Galerkin(LDG)method for fourth-order time-dependent problems in one-dimensional and multidimensional spaces.The fourth-order time-dependent problem is a kind of important diffusion problem,which has been widely used in engineering problems,such as the modeling of thin beams,strain gradient elasticity and phase separation in binary mixtures.Furthermore,The Local Discontinuous Galerkin method,which was developed out of the Discontinuous Galerkin(DG)method,has high order accuracy and efficiency in solving partial differential equations.However,the mathematical models of many practical problems,such as the problem of pure bending of beam,have more general boundary conditions.The novelty of this work is that the boundary condition is not the normal periodic boundary conditions but the Neumann boundary conditions.In this work,we analyze the convergence properties of the LDG method for fourth-order time-dependent problems with Neumann boundary conditions in one-dimensional and multidimensional spaces.First of all,we study the LDG method for fourth-order time-dependent problems with Neumann boundary conditions in one-dimension spaces.Error estimates results are given by choosing the proper numerical fluxs and projections.What’s more,some basic inequalities and the knowledge in approximation theory are also needed.To make the result more general,we extend the analysis to multidimensional space.At last,numerical experiments are displayed to verify the theoretical results,in which third order Runge-Kutta method is used for time discretizations.