| The convection-dominated convection-diffusion-reaction equations are an important class of singularly perturbed differential equations.These equations are widely used to model many problems in the fields of science,technology and practical engineering applications,such as fluid flows,water pollution,simulation of oil extraction from underground reservoirs,flows in chemical reactors,convective heat transport problems with large Peclet numbers,semiconductor device simulation and so on.Since the exact solutions of the equations contain sharp layers（boundary or inner layers）,the numerical solution obtained by traditional numerical scheme has non-physical oscillations near the sharp layers,and as the perturbation parameter becomes smaller,the oscillations will also propagate to the inner domain.Generally,there are two ways to construct stable numerical schemes overcoming numerical pseudo-oscillations.One is to continuously refine the grid,for example,the spatial grid parameters of the subdivision are far less than 1（h ≤ 1）.However,using too fine grid in practical calculation will not only occupy computer memory,but also waste a lot of CPU time,which increases the amount of computation by orders of magnitude and seriously affects the application of numerical methods in solving practical problems.Another approach is to try to establish non-standard methods that are different from standard numerical methods,such as characteristic finite element method,discontinuous finite element method,streamline upwind Petrov-Galerkin（SUPG）method,local projection stabilization（LPS）method,Galerkin least-squares（GLS）method,continuous interior penalty（CIP）method,etc.These methods avoid numerical pseudo-oscillations to some extent,especially have good resolution for the solutions of large gradient and large deformation problems,and are widely used in the numerical calculation of solid and fluid mechanics.At present,the researches on unsteady convection-diffusion-reaction equations mainly focus on the above non-standard stable finite element methods for spatial discretization and the finite difference schemes for temporal discretization,but the numerical schemes are difficult to obtain high accuracy in time direction.In order to overcome this shortcoming,some spatial stabilization techniques have been extended to the space-time finite element framework.Although the stabilized numerical schemes have high accuracy in space and time,the dimension of the approximate solution is increased due to the unified treatment for time and space variables,i.e.,when the space dimension is n,the n+ 1-dimensional problem needs to be considered.This spatial and temporal discretization needs to solve large-scale linear algebraic equations,which not only increases the complexity of the method,but also affects the application of numerical methods in solving practical problems.Nevertheless,at present,only a few kinds of spatial stabilization methods have been introduced into the space-time finite element methods.In this thesis,the space-time finite element method and the local projection stabilization method are combined to study the convection-diffusion-reaction equations with space variable coefficients and space-time variable coefficients.The space-time continuous and time-allowed discontinuous and space continuous stabilized space-time finite element schemes are established.This kinds of numerical schemes not only have high order accuracy in time and space,but also can effectively suppress pseudo-oscillations.Meanwhile,to overcome the disadvantage that space-time discretization increases the amount of calculation,we will use the special space-time finite element basis functions.According to the continuity or discontinuity in temporal direction of numerical schemes,the Lagrange interpolation polynomials based on special points are constructed for time,which can decouple time and space variables.This construction and analysis method can simplify the theoretical analysis and reduce the amount of actual calculation on the premise of ensuring the advantages of traditional space-time finite element method.The main results of this thesis are as follows:·The time discontinuous space-time finite element scheme and continuous spacetime finite element scheme based on local projection stabilization technique are constructed for the convection-diffusion-reaction equations with spatial variable coefficients,respectively.According to the different situations of continuity and discontinuity in temporal direction of numerical schemes,the Lagrange interpolation polynomials determined at special points（discontinuous: Radau points,continuous: Legendre points and Lobatto points）and the corresponding Gauss-type numerical integration formulas（discontinuous:Gauss-Radau numerical integration,continuous: Gauss-Legendre numerical integration and Gauss-Lobatto numerical integration）are introduced.By using the properties of interpolation polynomials and finite element techniques and the high algebraic accuracy characteristics of Gauss-type numerical integration,the well-posedness analysis and a priori error estimate of the stabilized space-time finite element schemes are given.It is proved that the error estimate of the finite element solution in L∞（L2）-norm is O（hr+s+1）using the elliptic projection operator,where h and srepresent the spatial grid parameter and time step,r and represent the polynomial degree selected in the spatial and temporal directions,respectively.Further,a special interpolation operator satisfying the standard approximation and additional orthogonality is constructed in the spatial direction,and the Ritz projection operator based on the stable bilinear form is introduced,which can not only simplify the process of theoretical analysis,but also improve the corresponding error estimate result to O（（ε1/2+ h1/2）hr+ s+1）,where ε is the perturbation parameter.·The local projection stabilization time discontinuous space-time finite element method is extended and applied to the convection-diffusion-reaction equations with spacetime variable coefficients.Since the coefficients of the equations depend on time variable,the Gauss-type numerical integration formula will no longer be accurate,so the theoretical analysis and algorithm implementation are more difficult than the equations with spatial variable coefficients.In error estimate,if the previous processing method is adopted,it is difficult to obtain high order spatial estimate results for the numerical integration remainder generated by the convection term,which results in the reduction of the overall convergence order of the spatial direction.Therefore,we construct the space-time finite element function based on the special interpolation operator in the spatial direction,and make full use of the approximation and orthogonality of the operator in the error analysis to obtain the high order convergence results in space.·For the stabilized time discontinuous space-time finite element scheme and continuous space-time finite element scheme established in this thesis,the corresponding numerical examples are given to verify the correctness of the theoretical analysis and the validity and operability of the stabilized numerical scheme.Furthermore,two kinds of stabilized space-time element schemes are compared in terms of computational accuracy and computational complexity.The influences of stabilization parameters on the finite element scheme and the superconvergence phenomenon are discussed. |
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