Recovery-based Error Estimator For The Natural-convection Problem

Natural convection?NC?problems often occur in practical problems.In atmo-spheric dynamics,it is an important forced dissipative nonlinear system.It contains not only the velocity vector field and the pressure field,but also the temperature field.From the thermodynamic point of view,we know that the flow viscosity of the fluid will produce certain heat,therefore,fluid Movement must With the mutual conversion of temperature,velocity and pressure,the study of this nonlinear system is of great significance.First,the speed u and pressure p are restricted by the incompressible condition?·u=0,the problem is more difficult.Therefore,we propose a quasi compressible finite element method to solve this problem.Existence and uniqueness,The discrete velocity space and pressure space must satisfy the discrete LBB condition.Secondly,we propose a method based on the loss correction to solve the NC problem.The main idea of the defect correction method is that we solve the stable nonlinear problem of the artificial viscosity coefficient at a loss step,and correct the residual by the linearization equation at the school.We all use Oceen iteration.As we all know,the loss correction method is an iterative improvement technique for solving nonlinear steady state problems.This method improves the accuracy of the understanding and is effective for the convection dominated problem without the need of the grid.For the NC equation,we consider the stress tensor?=?u-pI+?T.If the true solution is enough regular,then it can ensure that the stress tensor is a continuous space.We use the lower-order finite element pair for the P₁-P₀-P₁discrete NC problem.The finite element approximation of the stress tensor?is a discontinuous piecewise constant for the?_h=?u_h-p_hI+?T_h.We use the superconvergent slice recovery technique to construct continuity.The amount of space is G??_h?,and the local restorative error estimator||?_h-G??_h?||_Kis obtained.This estimator is a calculable scalar which can reflect the local error information.It can provide the finite element approximation error size and the information in the whole region.It can automatically control the finite element mesh generation and the subdivision of the finite element.The numerical algorithm is selected to ensure the effectiveness and flexibility of the finite element method.In this paper,this restoration error estimator is used in the penalty finite element method and the loss correction method,and the consistent numerical results are obtained.