Stability Analysis Of Weak Galerkin Finite Element Methods For Elliptic Problems

In this paper, we focus on the stability of weak Galerkin finite element methods in the weak function space. Considering homogeneous boundary value problems of elliptic equation where Ω is a bounded domain in R". When f ∈ L2, the stability of the following is proved ‖u‖2≤C‖f‖0,f∈L2. Especially, if f ∈ H-1,the ‖u‖1≤C‖f‖-1.In standard finite element theory, we have following variational form of the elliptic problems:seek u e ∈H10(Ω), satisfying where a(u, v)= (▽u, ▽v). Using the positive definiteness of the bilinear form a(u, v), it is not difficult to see that (?)f∈L2,‖uh‖≤1/y‖f‖0.Further more, when f ∈ H-1,variational form can be represented as:seek u ∈ H10(Ω), satisfying where 〈.,.〉 is dual pair. Similarly, we can get stability that‖uh‖1≤C‖f‖-1.In the current study on the weak Galerkin finite element methods,‖|·‖| car be defined in weak finite element space by the bilinear form. Actually,‖|·‖| and ‖·‖1 are the corresponding norm, but not equivalent. We hope that weak Galerkin equations can be applied to a wider range of functions. To solve this problem, we hope to define norm‖|·‖|-1 corresponding to ‖·‖-1, and use this norm to discuss the stability of the weak Galerkin finite element methods in this paper. In other words, we will prove the following stability estimate‖|uh‖|≤C‖|f‖-1, where f with less smoothness.In the configuration of second order elliptic equations and fourth order Biharmonic equation in weak Galerkin method, by approximate analysis and calculation illustrates the weak Galerkin method configuration, we illustrate the necessary of weak stabilizer and the construction principle. Such organically combined the weak Galerkin approximate equation and original equation and finite element method.