| In this paper,we consider the numerical method of a forth-order parabolic equation through the weak Galerkin finite element.And the initial-boundary value problem is follow-ing ut+Δ2u=f, x∈Ω,0≤t≤t, (0.1) u=au/an=0, x∈aΩ,0≤t≤t, (0.2) u(·,0)= Ψ,x∈Ω. (0.3) where A is the Laplace operator and Ω is open bounded in Rd (d=2,3) with Lipschitz-continuous boundary aΩ.Let H= L2(Ω) with standard inner product (·,·) and norm ||·||.We also denote by Hm=Hm(Ω) the standard Sobolev space and Then,for the equation (0.1)-(0.3),a natural variational formulation is to find u ∈ L2(0,t;H02(Ω)) satisfying u(0,·)=Ψ such that (ut, v)+(Δu,Δv)=(f, v), (?)v ∈ H02(Ω). (0.4)The purpose of the present paper is to consider a newly developed weak Galerkin (WG) finite element method for the equation (0.1)-(0.3). The basic idea of WG finite element method is to use some weak function space W(Ω) to approximate H2 and define a weak Laplace operator Δw instead of the standard Laplace operator Δ.A weak continuity can be realized by introducing an appropriately defined stabilizer,denoted as s(·,·).Let Th, be a quasiuniform family of triangulations partition of the domain Ω.T be each triangle element with mesh size hT,and h=max{hT}T∈T. For any given non-negative integer k≥2,denote by Pk(T) the set of polynomials on T with degree no more than k,denote by Pk(e) the set of polynomials on e (?) aT with degree no more than k,and denote by Wk(T) the discrete weak function space given by Wk(T)={{v0,vb,vg}:v0∈Pk(T),vb∈Pk(e),vg∈[Pk-1(e)]d,e(?)aT}. then we get a weak finite element space Vh Vh={{v0,vb,vg:{v0,vb,vg}|T∈Wk(T),(?)T∈Th}. Denote by Vh0 the subspace of Vh with vanishing traces;i.e., Vh0={v={v0,vb,vg}∈Vh,vb|e=0,vg·n|e=0,e(?)aT ∩aΩ}. For any uh={u0,ub,ug} and v={v0,vb,vg} in Vh,we introduce a bilinear form s(uh,v) A discrete weak Laplace operator,denote by △w is defined as the unique polynomial △w∈ Pk(T)satisfying the following equation (△wv,φ)T=(v0,△φ)T-<vb,▽φ·n>aT+<vg·n,φ>aT,(?)φ∈Pk(T). We also introduce the following notation Formally,our weak Galerkin finite element method for(0.1)-(0.3)can be described by seek-ing a finite element function uh∈L∞(0,t;Vh)such that (vh,t(t,·),v0)+(△wuh,Δwv)h+s(uh,v)=(f,v0),(?)v={v0,vb,vg}∈Vh0.(0.5)For some discrete time weak Galerkin method,we introduce a time step K and tn= nk,n=1,…,N,where Nk=t. And denote by Un∈Vh the approximation of u(tn) to be determined.The backward Euler Weak Galerkin method is to seek Un∈Vh(n≥ 1),satisfying U0=QhΨ such that (aUn,v0)+(△wUn,△wv)+s(Un,v)=(f(tn),v0),(?)v∈Vh0. (0.6)And then,we get some error analysis of continuous time and discrete time weak Galerkin methods.定理0.3.Let uh={u0,ub,ug} be weak Galerkin finite element solution arising from (0.5) with finite ement functions of order k≥2.Assume that the exact solution of (0.1)-(0.3) is sufficiently regular such thaf u∈Hmax{k+1,4}(Ω).Dentote by eh=uh-Qhu the difference between the weak Galerkin approximation and the L2 projection of the exact solution u.The there exists a constant C such that||eh||2+(?)01|||eh(·,t|||2dτ≤||eh(·,0)||2+Ch2k-2((?)01(||u||k=12+h2δμ,2||u||42ds),(0.7) and 4(?)01||eh,τ||2dτ+|||eh|||2 ≤2|||eh(0,·)|||2+Ch2k-2(||u||k+12+||u(0,·)||k+12+hδk,2(||u||42+||u(0,·)||42) (0.8) +(?)(""uτ||k+12+hδk,2||uτ||42)dτ+(?)0(||u||k+12+hδk,2||u42)dτ).定理0.4.Let u ∈Hmax{1+k,4} (Ω) and Un be the solutions of (0.1)-(0.3) and (0.6),re-spectively.Denote by ehn=Un-Qhu(tn) the difference between the ackward Euler weak Galerkin approximation and the L2 projection of the exect solution u.Then there exist a constant C such that... |
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