Finite Element Analysis For Coupled Nonlinear Partial Differential Equations

This thesis is devoted to the error analysis of finite element methods for non-linear coupled partial differential equations?such Thermistor equation?Poisson-Nernst-Planck?PNP?equation?Navier-Stokes equation?with semi-discrete and fully-discrete schemes.The convergence?superclose and superconvergence prop-erties are investigated in view of conforming finite element method?nonconforming finite element method and mixed finite element method.The main innovations of this thesis are as follows:?1?Different from optimal error estimates in the existing literature,by skill-fully using the integral identities technique of low order bilinear element and ex-tended rotated₁element?₁?on the rectangular mesh,the difficulty caused by the square of the gradient of the electric potentialin the coupled term??|?|²is overcame.Thus,the superclose and superconvergence results for the temperatureand electric potentialare obtained.?2?For the PNP equation,since the gradient of the electrostatic potentialappeared in the coupled terms?₁??and?₂??.Only the sub-optimal error estimates in²-norm of the mass concentration of particles₁and₂are derived in existing literature.Here,contrary to existing results,in terms of the high accu-racy result of bilinear element on the rectangular mesh,not only the sub-optimal error estimates in²-norm of the mass concentration of particles are improved,but also the superclose and superconvergence results are derived.?3?For the Navier-Stokes equations,by employing the special character of a low order nonconforming mixed element,i.e.,the velocity approximated by the constrained rotated₁?CNR₁?element and the pressure approximated by the piecewise constant?₀?element,and using the skillful estimate of the convection term,the superlose and superconvergence error estimate in broken¹-norm for velocityand in²-norm for pressureare achieved.?4?Furthermore,for the Navier-Stokes equations,by using the low-order conforming mixed element,i.e,the velocity approximated by the bilinear element₁₁and the pressure approximated by the piecewise constant₀element,under a weaker assumption to the domain?for example the boundary belongs to²?compared with the existed literatures,by using the error splitting technique,the corresponding optimal error estimates of velocityand pressureare obtained without any restrictions on the temporal step size and the spatial step size.In the first part,time-dependent nonlinear Thermistor equation?also called Joule heating equation?is studied by using bilinear element with semi-discrete and linearized backward Euler fully-discrete schemes,and the superclose and superconvergence results are attained.Since|?|²appeared in the coupled ter-m,optimal error estimate for temperatureand electric potentialwere ob-tained.Contrary to the traditional way,by skillfully using the integral identities of bilinear element on the rectangular mesh,and the mean value technique on element,the difficulty caused by the square of the gradient is overcame.Thus,the superclose result of the corresponding variables are achieved.Based on this achievement,by a simple and effective interpolation postprocessing operator,the global superconvergence results are acquired.In the second part,a popular low order nonconforming element,i.e.,extended rotated₁?₁?element,is applied to the Thermistor equation,and the su-perclose and superconvergence results are established with the semi-discrete and linearized backward Euler fully-discrete schemes.Compared to the conforming ones,the consistency error always appeared in the error analysis for the noncon-forming elements,and it is difficult to get high order result in energy norm.With the help of the two special properties:One is the interpolation operator is equiv-alent to the Ritz projection operator;the other is the consistency error is of order??²?,which is one order higher than its interpolation error,and using the mean value on element,the superclose results in energy norm are obtained.Further-more,the global superconvergence results are obtained by a proper interpolation postprocessing operator.In the third part,the time-dependent PNP equation is discussed by us-ing bilinear element with the semi-discrete and fully-discrete schemes.Since the gradient of the electrostatic potential appeared in the coupled term,only the sub-optimal error estimates were obtained in the existing literature.Different from the traditional approach,in terms of the special property of bilinear ele-ment on rectangular mesh,i.e.,the high accuracy result?see the first part?,the difficulty caused by the gradient in the coupled term is overcame.Not only the sub-optimal error estimates in²-norm of the mass concentration of particles are improved,but also the superclose results in energy norm are achieved.By using the same interpolation postprocessing operator used in the first part,the global superconvergence results are got.In the fourth part,a low order nonconforming mixed element,i.e.,the velocity approximated by the constrained rotated₁?CNR₁?element and the pressure approximated by the piecewise constant?₀?element,is employed to study the time-dependent Navier-Stokes equation with linearized fully-discrete scheme.In terms of the high accuracy results of the above element pair,by introducing local²projection and using the special splitting technique of the convection term,the superclose estimates in energy norm for velocityand²-norm for pressureare obtained.Furthermore,the global superconvergence results are acquired by constructing proper interpolation postrocessing operators.In the last part,a low order conforming mixed element,i.e,the velocity approximated by the bilinear element₁₁and the pressure approximated by the piecewise constant₀element,is applied to discussed the time-dependent Navier-Stokes equation with linearized fully-discrete scheme.With the help of the error splitting technique,under a weaker smooth assumption to the domain,the corresponding optimal error estimates of the velocityand the pressureare attained without any restrictions on the temporal step size and the spatial step size,which reduces the requirement that boundary belongs to²in the existing literature.It is worth mentioning that we provided the corresponding numerical exam-ples for the each part above to demonstrate the correctness of the theoretical analysis and the efficiency of the presented method.