Research On New Finite Element Methods For Evolutionary Partial Differential Equations

The finite element method(FEM)is an efficient and widely used numerical method for computing solutions of partial differential equations(PDEs)arising from science and engineering.The method is largely based on variational principles for the underlying PDEs,with a numerical procedure that approximates analytical func-tions by piecewise polynomials.The algorithmic and mathematical formulation of FEM involves the construction of“elements”(i.e.,specific polynomial functions)associated with a partition of the physical domain,and a strategy on how these el-ements might be used to localize global variational problems into element-wise local variational problems.FEM is generally preferred and often ideal for solving partial differential equations on domains with complex geometry.Since its first discovery in the middle of the last century,FEM has quickly become a useful and indispensable computational tool in scientific computing and research.Many finite element code packages have been developed for solving challenging PDE problems from practical applications.With the rapid advance of science and technology,the development of new finite element methods that are robust and efficient for complex modeling prob-lems is still one of the most important and challenging research areas in science.This thesis is in response to this need and shall focus on the development of new finite element methods for important evolutionary partial differential equations.Our first new result is an over-penalized weak Galerkin(OPWG)numerical method for the parabolic problem.Following the approaches of the classical discon-tinuous Galerkin and weak Galerkin finite element methods,the OPWG numerical scheme is formulated by assuming double-valued weak functions on interior edges of the finite element partition in the usual weak Galerkin formulations.From the double-valued functions defined on interior edges,it is natural to generate over-penalized jump terms as part of the stabilization,which then simplifies the computation signif-icantly.In this research,we first develop a semi-discrete stable scheme,and then a fully-discrete one with implicit ?-algorithm in time for 1/2???1,including first-order backward Euler(?=1)and second-order Crank-Nicolson(?=1/2).Next,for parabolic interface problems with diffusion coefficients varying in both temporal and spatial directions,we shall devise semi-discrete and fully-discrete numerical schemes using backward Euler in time.We show that the semi-discrete numerical scheme is un-conditionally stable.An optimal order of error estimate in the energy norm is derived for both the semi-discrete and the fully-discrete schemes through the establishment of some error equations.By virtue of an elliptic projection operator,some optimal order error estimates in L~2 norm are derived.Numerical experiments are conducted and presented to validate the e ciency and the order of convergence predicted in theory.Another evolutionary PDE studied in the research is the Biot's consolidation model.Among many applications,the Biot's consolidation model can be used to describe the flow of fluid in an elastic porous media.Our research involves a re-formulation of the Biot's model equations through the introduction of a new variable called“total stress”.A new three-field finite element method is devised by using the total stress as an unknown,which couples the displacement and fluid pressure vari-ables.The three fields include displacements,total stress,and fluid pressure.Optimal order error estimates are derived for the semi-discrete and fully-discrete numerical schemes for the Biot's consolidation model.The fully-discrete numerical scheme is based on the backward Euler in time.The fully-discrete scheme is implemented for some benchmark examples,and the computational results show a great consisten-cy between theory and computation for two finite element methods associated with the elastic part of the modeling system.We further develop a four-field mixed fi-nite element method for the Biot's consolidation model,where the four fields include the displacement,the total stress,the flux and the pressure for the porous medium component of the modeling system.The mixed finite element method makes use of the Raviart-Thomas element for the porous medium flow equation,while the Crank-Nicolson scheme is employed for the time discretization.The main contribution of this work is the derivation of an optimal order error estimate for the semi-discrete and the fully-discrete schemes in energy or L~2 norms for appropriate approximating functions.Numerical results are presented to validate the theoretical error estimates.Our next evolutionary PDE involves fractional order differential operators in s-patial direction.This research addresses solution techniques for time-dependent frac-tional order PDEs by using multigrid methods for large linear systems arising from finite element discretizations.This work established a fast convergence for the V-cycle multigrid FEM for time-dependent fractional order problems that is uniform in terms of a parameter ?,even when ??0.Numerical experiments are performed to ver-ify the convergence with O(NlogN)complexity by using the fast Fourier transform method.In the study of virus transmission related to COVID-19,we begin to investigate numerical methods for coupled Navier-Stokes and linear transport equations.Our first numerical method is a new weak Galerkin finite element method for the steady-state Stokes equation.This new weak Galerkin method is devised by introducing a new formula for the usual weak gradient that permits the use of polynomials of arbitrary order for all the variables involved.Optimal order error estimates for the velocity approximation in energy norm and L~2 norm are established,while the error estimate for the pressure approximation is derived in L~2 norm.Various numerical experiments are conducted to illustrate the e ciency,accuracy,and stability of the new weak Galerkin method.A further development for time evolutionary Navier-Stokes equation coupled with the linear transport equation is anticipated as a follow up of this thesis work.