Numerical Theory And Calculation Of Two Kinds Of Fluid Equations By The Over-penalized Weak Galerkin Methods

In this work,an over-penalized weak Galerkin method(OPWG)is used to study two types of fluid equations.This method is an extension of the weak Galerkin method that was first introduced by Wang and Ye.Different from the latter,the former is double-valued on the interior edges of elements.Naturally,one can introduce an over-penalized term similar to the interior penalty discontinuous Galerkin methods.OPWG makes the selection of basis functions more flexible while optimal penalty parameters can be obtained.It is more convenient to deal with some problems with discontinuous or low regular solutions,and superconvergence results can also be proved in some cases.The first part of the thesis lies in approximating the steady-state convection-diffus-ion equations by OPWG.Here,the polynomial approximation space is first taken as(P_k,P_k,[P_k-1]²),then we introduce a discrete weak gradient operator and discrete weak convection operator.With a careful deduction,we establish the error equations,and then prove optimal convergence rates in the energy norm and the L²norm of nu-merical solution.To the numerical examples,we first present errors and a selection of the optimal penalty parameters of the OPWG scheme for given different coefficients,and make a comparision among different penalty parameters.Numerically,the same optimal convergence orders can be obtained as the theoretical results.The second part of the thesis mainly concerns about numerical analysis of the steady-state incompressible Navier-Stokes equations approximated by OPWG,we use a vectorial polynomial space(P_k²,P_k²,[P_k-1]^2×2)to approximate the velocity and apply a piecewise polynomial space(Pk-1)to approximate the pressure function.A suitable inf-sup condition under the approximation spaces can be established.Therefore,we can prove the existence and uniqueness of the OPWG solution to approximate the Navier-Stokes equation.Theoretically,the optimal order error estimate for the velocity in the energy norm and the pressure in the L²norm can be derived from their corresponding error equations.