Convergent Adaptive Least-Squares Finite Element Methods For Linear Convection Equation

The first order hyperbolic problem is a kind of equation which is widely used in engineering physics, on the problem of numerical calculations, the researchers mainly use discontinuous finite element method to solve the problem, and Zhang Tie et al. analyzed this method with a posteriori error, This method have advantage for solving the problem with dramatic change problem, but discontinuous finite element is more complicated respect to the continuous finite element and is not easy to construct convergence adaptive algorithm.With the development of large-scale scientific computing and parallel computing environments, domain decomposition and adaptive iterative solver has been a wide range of applications in the numerical solution of partial differential equations. The adaptive grid technology is global solution the problem, the method have many advantages that other methods do not have. First, in the local area only to solve the small size equation group, the error of the exact solution can be obtained for the local area, the complexity is relatively small; second, adaptive methods choose main error grid cell to encrypted, and iterations, this method is suitable for computer development and the eventually grid is based on the final results of solving regional characteristics, the result can be evenly approximation error distribution and control error convergence order.In this paper, the characteristic of the least square method is used to estimate the posterior error of the first order hyperbolic problem, which makes the error estimation has a good characteristic of Galerkin orthogonality. Least squares form likeswe got an error for solution the space problems in discrete finite element space in each unit, so in the region we using adaptive meshing techniques, under the control of the entire region, we solve the case of error order to obtain an adaptive convergence least squares finite element for solving first order hyperbolic theory and proved then reasonable controllability can be adaptive mesh to get the numerical solution, finally, we use practical examples to verify the theory demonstrated herein correctness. The solution method is geometric convergence, and the convergence factor can be adjusted to achieve optimal adaptive according to the convergence effect desired.In this paper, the convergence of the adaptive method is based on the continuous finite element, can achieve better for one order hyperbolic problems are solved, can adapt to different forms of practical engineering problem solving, and the theory have a strong promotional value.