An Inf-Sup Stabilized Finite Element Method by Multiscale Functions for the Stokes Equations

In the paper,an inf-sup stabilized finite element method by multiscale functions for the Stokes equations is discussed.The key idea is to use a PetrovGalerkin approach based on the enrichment of the standard polynomial space for the velocity component with multiscale functions.The inf-sup condition for P_(1)-P_(0)triangular element(or Q_(1)-P_(0)quadrilateral element)is established.The optimal error estimates of the stabilized finite element method for the Stokes equations are obtained.

Multiscale Basis Functions for Singular Perturbation on Adaptively Graded Meshes

We apply the multiscale basis functions for the singularly perturbed reaction-diffusion problem on adaptively graded meshes,which can provide a good balance between the numerical accuracy and computational cost.The multiscale space is built through standard finite element basis functions enriched with multiscale basis functions.The multiscale basis functions have abilities to capture originally perturbed information in the local problem,as a result our method is capable of reducing the boundary layer errors remarkably on graded meshes,where the layer-adapted meshes are generated by a given parameter.Through numerical experiments we demonstrate that the multiscale method can acquire second order convergence in the L^(2)norm and first order convergence in the energy norm on graded meshes,which is independent ofε.In contrast with the conventional methods,our method is much more accurate and effective.

Numerical Investigation on the Boundary Conditions for the Multiscale Base Functions

We study the multiscale finite element method for solving multiscale elliptic problems with highly oscillating coefficients,which is designed to accurately capture the large scale behaviors of the solution without resolving the small scale characters.The key idea is to construct the multiscale base functions in the local partial differential equation with proper boundary conditions.The boundary conditions are chosen to extract more accurate boundary information in the local problem.We consider periodic and non-periodic coefficients with linear and oscillatory boundary conditions for the base functions.Numerical examples will be provided to demonstrate the effectiveness of the proposed multiscale finite element method.

A Sub-Grid Structure Enhanced Discontinuous Galerkin Method for Multiscale Diffusion and Convection-Diffusion Problems

In this paper,we present an efficient computational methodology for diffusion and convection-diffusion problems in highly heterogeneous media as well as convection-dominated diffusion problem.It is well known that the numerical computation for these problems requires a significant amount of computermemory and time.Nevertheless,the solutions to these problems typically contain a coarse component,which is usually the quantity of interest and can be represented with a small number of degrees of freedom.There are many methods that aim at the computation of the coarse component without resolving the full details of the solution.Our proposed method falls into the framework of interior penalty discontinuous Galerkin method,which is proved to be an effective and accurate class of methods for numerical solutions of partial differential equations.A distinctive feature of our method is that the solution space contains two components,namely a coarse space that gives a polynomial approximation to the coarse component in the traditional way and a multiscale space which contains sub-grid structures of the solution and is essential to the computation of the coarse component.In addition,stability of the method is proved.The numerical results indicate that the method can accurately capture the coarse behavior of the solution for problems in highly heterogeneous media as well as boundary and internal layers for convection-dominated problems.