A Study Of Discontinuous Galerkin Method And Neural Network Method For Diffusion Equations

Diffusion equations have many applications in engineering research,such as electromagnetic fields,heat conduction,fluid mechanics,etc.Most diffusion equations cannot be solved theoretically,so it is important to develop reasonable numerical methods to find their numerical solutions.In this thesis,we research a group of steady-state interface diffusion equations,namely elliptic interface problem,and a group of time-dependent diffusion equations.In chapter 3,we develop a direct discontinuous Galerkin method to solve elliptic interface problems with unfitted mesh.The interface is a complex geometry in which the effort of dissection can be greatly reduced by using unfitted mesh.In this chapter,the interface problem is solved by direct discontinuous Galerkin.For the non-interface element,every element is approximated by a general p degree polynomial.For the interface element,which is partitioned by the interface into two independent parts,every part has at least one sub-domain internal non-interface element which is not empty of its interaction.The two parts of the interface element are approximated by the p degree polynomial of the two internal elements.The global discrete variational problem is given to demonstrate the well-posedness and the error estimates of the numerical solution.The numerical solution is achieving the optimal order in respect to both the energy norm and L2 norm error estimate.The validity of the method and the correctness of the theoretical analysis are proved by the numerical examples.In chapter 4,we develop a cell-average based neural network method for solving timedependent high-dimensional diffusion equations.The traditional numerical methods for solving high-dimensional diffusion equations must require the strict CFL condition as the spatial mesh is refined and the time mesh size also needs to be refined.In this chapter,the CANN method which we develop could avoid the CFL condition in solving high-dimensional problems.The CANN method is based on a weak format for integrating differential equations.Initial values at t=t0 and approximate solution at t=t0+Δt obtained by high order numerical method are taken as the inputs and outputs of the network,respectively.We use supervised training combined with a simple backpropagation algorithm to train the network parameters.The trained network can be used to solve the same high-dimensional equation with different initial values.And we could get the network-approximated solutions of different times t=t0+n×Δt,where n=1,2,….The numerical examples also show the error is only related to the spatial size.With spatial mesh subdivision,the CANN method has remarkable accuracy for large time step size.