A NEW FINITE ELEMENT SPACE FOR EXPANDED MIXED FINITE ELEMENT METHOD

In this article,we propose a new finite element spaceΛh for the expanded mixed finite element method(EMFEM)for second-order elliptic problems to guarantee its computing capability and reduce the computation cost.The new finite element spaceΛh is designed in such a way that the strong requirement V h⊂Λh in[9]is weakened to{v h∈V h;d i v v h=0}⊂Λh so that it needs fewer degrees of freedom than its classical counterpart.Furthermore,the newΛh coupled with the Raviart-Thomas space satisfies the inf-sup condition,which is crucial to the computation of mixed methods for its close relation to the behavior of the smallest nonzero eigenvalue of the stiff matrix,and thus the existence,uniqueness and optimal approximate capability of the EMFEM solution are proved for rectangular partitions in R d,d=2,3 and for triangular partitions in R 2.Also,the solvability of the EMFEM for triangular partition in R 3 can be directly proved without the inf-sup condition.Numerical experiments are conducted to confirm these theoretical findings.

Application of the expanded distinct element method for the study of crack growth in rock-like materials under uniaxial compression

The expanded distinct element method(EDEM)was used to investigate the crack growth in rock-like materials under uniaxial compression.The tensile-shear failure criterion and the Griffith failure criterion were implanted into the EDEM to determine the initiation and propagation of pre-existing cracks,respectively.Uniaxial compression experiments were also performed with the artificial rock-like samples to verify the validity of the EDEM.Simulation results indicated that the EDEM model with the tensile-shear failure criterion has strong capabilities for modeling the growth of pre-existing cracks,and model results have strong agreement with the failure and mechanical properties of experimental samples.The EDEM model with the Griffith failure criterion can only simulate the splitting failure of samples due to tensile stresses and is incapable of providing a comprehensive interpretation for the overall failure of rock masses.Research results demonstrated that sample failure primarily resulted from the growth of single cracks(in the form of tensile wing cracks and shear secondary cracks)and the coalescence of two cracks due to the growth of wing cracks in the rock bridge zone.Additionally,the inclination angle of the pre-existing crack clearly influences the final failure pattern of the samples.

Analysis of Two-Grid Methods for Nonlinear Parabolic Equations by Expanded Mixed Finite Element Methods

In this paper,we present an efficient method of two-grid scheme for the approximation of two-dimensional nonlinear parabolic equations using an expanded mixed finite element method.We use two Newton iterations on the fine grid in our methods.Firstly,we solve an original nonlinear problem on the coarse nonlinear grid,then we use Newton iterations on the fine grid twice.The two-grid idea is from Xu's work[SIAM J.Numer.Anal.,33(1996),pp.1759–1777]on standard finite method.We also obtain the error estimates for the algorithms of the two-grid method.It is shown that the algorithm achieve asymptotically optimal approximation rate with the two-grid methods as long as the mesh sizes satisfy h=O(H^((4k+1)/(k+1))).

A Two-grid Method with Expanded Mixed Element for Nonlinear Reaction-diffusion Equations

Expanded mixed finite element approximation of nonlinear reaction-diffusion equations is discussed. The equations considered here are used to model the hydrologic and bio-geochemical phenomena. To linearize the mixed-method equations, we use a two-grid method involving a small nonlinear system on a coarse gird of size H and a linear system on a fine grid of size h. Error estimates are derived which demonstrate that the error is O（△t ＋ h k＋1 ＋ H 2k＋2 d/2 ） （k ≥ 1）, where k is the degree of the approximating space for the primary variable and d is the spatial dimension. The above estimates are useful for determining an appropriate H for the coarse grid problems.