Weak Galerkin Finite Element Method For Second Order Parabolic Equations

Parabolic partial differential equations are used to describe a wide family of sci-entific problems. For example, the mathematical models of heat conduction, diffusing phenomenon and biology are often represented by parabolic partial differential equation-s. At present, there exist many classical numerical methods for parabolic equations, including finite difference methods, finite element methods, finite volume methods and so on.In this paper, we mainly study two different kinds of weak Galerkin finite methods for initial-boundary value problems of the second order parabolic equations. We investi-gate the stability, energy conservation and the error estimations of numerical solutions. Moreover, numerical examples are used to verify the theoretical results. This paper is divided into three parts. In the first part, we first state classical numerical methods of the second order parabolic equations. Then, we briefly recall some related results about weak Galerkin finite element methods. In the second part, we establish (r, r, r-1)-order weak Galerkin finite element schemes and provide the stability, energy conservation and the convergence estimations. In the third part, we propose (r, r-1, r-1)-order weak Galerkin finite element methods. We further analyse the stability and energy preserving property, and show the convergence and the optimal order estimation of semi-discrete and fully discrete schemes.In Chapter one, we first state the actual backgrounds of parabolic problems. Sec-ondly, we recall some related results about numerical methods obtained by authors both at home and abroad for parabolic equations in recent years. Then, we introduce the clas-sical derivative, Sobolev space, and the classical finite element methods for second order parabolic equations. At last, we describe the definition of weak gradient, weak Galerkin finite element methods for parabolic problems developed by Q. H. LI and J. WANG [18].In Chapter two, We consider the following initial-boundary value problem for the second order parabolic equation where Ω is a polygonal domain in R2, J= (0, t0] with t0> 0, a = a(·)2×2 ∈ [L∞(Ω)]2×2 is a symmetric matrix-valued function. Assume that the matrix function a(-) satisfies the following property:there exist two constants 0<α1<α2 such thatLet Th be a family of partitions of the domain Ω, where h is the partition diameter. We assume throughout the paper that Th is some shape regular A1-A4 [24,38]. Denote by T0 its interior and by (?)T its boundary for any T ∈ Th, respectively. For each T ∈Th, r≥1, let Pτ(T0) and Pτ((?)T) be the set of polynomials on T0 and on (?)T, respectively, with degree no more than r. Denoted by Vh the weak function space on Th given by and For T∈Th, define the local weak gradient operator ▽d from Vh to Gτ-1(T):= [Pτ-1(T)]2, which is determined by Define two bilinear forms on Vh,:for any v, w ∈ Vh, Denote by as(·,·) a stabilization of a(·,·) given by as(v, w)= a(v, w)+s(v, w). It is proved that there exist two constants α,β> 0 such that for u,v∈Vh, whereWe propose the continuous time weak Galerkin finite element method for the parabolic problem (8), based the weak Galerkin operator (9). The semi-discrete weak Galerkin finite element method is to find uh(t)={u0(·, t),ub(·,t)} ∈Vh0 for t≥0 such that uh(0)= Qhψ and the following equation holdsLet k> 0 be a time step-size. At the time level t= tn= nk, with integer 0< n< N, Nk= t0, denote by Un= Uhn={U0n, Ubn}∈Vh the approximation of u(tn). We further discretize above semi-discrete equation (10) with respect to t by the backward Euler method to obtain a full discrete weak Galerkin finite element method:seek Un ∈ Vh(n= 0,1,2,…, N) such that U0= Qhψ andwhere (?)Un= (Un-Un-1)/k.Then, we study the stability, energy conservation and convergence of the numerical algorithms (10) and (11) of the parabolic problem (8), and present numerical examples. Our main results are as follows:Theorem 1. Let uh(t)={u0(·, t),ub(·,t)} be the solution of the semi-discrete weak Galerkin finite element method (10) for the parabolic problem (8). Then there exists a constant independent of the mesh size h C> 0, such that i.e., the numerical solution is stable with respect to initial approximate value and forcing term.Theorem 2. Let uh(t)={u0(·, t), ub(·, t)} be the solution of the semi-discrete weak Galerkin finite element method (10) for the parabolic problem. Then uh(t) has the following energy preserving property on each T∈Th, i.e., for every T∈Th where and denote by Rh the L2 projection of [L2(T)]2 onto Gr-1(T).Theorem 3. Let u∈Hr+1(Ω) and uh be the solutions of parabolic problem (8) and the semi-discrete weak Galerkin finite element method (10) for the parabolic problem, respectively. Denote by e:=uh-Qhu ∈ Vh0 the difference between the weak Galerkin approximation and the L2 projection of the exact solution u. Assume u∈Hr+1(Ω). Then there exists a constant C> 0 independent of the mesh size h such that the following estimates hold andTheorem 4. Let u∈Hr+1(Ω) and Un be the solutions of the parabolic problem (8) and the full discrete weak Galerkin finite element method (11) for the parabolic problem , respectively. Denote by en:= Un-Qh’u(tn) the difference between the backward Euler weak Galerkin approximation and the L2 projection of the exact solution u. Assume u∈C2([0,t0];Hr+1(Ω)). Then there exists a constant C> 0 independent of the mesh size h such that for 0< n≤N and whereTheorem 5. Let u∈Hr+1(Q) and the corresponding elliptic problem has the H2-regularity. Then there exists a constant C> 0 independent of the mesh size h such that andTheorem 6. Let u ∈ Hr+1(Ω). Then there exists a constant C> 0 independent of the mesh size h such that andIn Chapter three, We consider the other weak Galerkin finite element method for the initial-boundary value problem (8). For each T∈Th, r≥1, let Pr(T0) be the set of polynomials on T0 with degree no more than r and Pr-1((?)T) be the set of polynomials on (?)T with degree no more than r - 1. Define weak Galerkin finite element spaces and For T∈Th, define the local weak gradient operator Vd from Vh to Gr-1(T):= [Pr-1(T)]2, which is determined by Define two bilinear forms on Vh:for any v,w∈Vh, Denote by as(·,·) a stabilization of α(·,·) given by as(v, w)= a(v, w)+s(v,w). It is proved that there exist two constants α,β> 0 such that for u,v∈Vh, whereWe propose the continuous time weak Galerkin finite element method for the parabolic problem, based the weak Galerkin operator (12). The semi-discrete weak Galerkin finite element method is to find uh(t)={u0(·,t),ub(·,t)}∈Vh0 for t≥0 such that uh(0)= Qhψ and the following equation holds Let k> 0 be a time step-size. At the time level t= tn= nk, with integer 0≤n≤ N,Nk= t0, denote by Un=Uhn={U0n,Ubn}∈Vh0 the approximation of u(tn). We further discretize above semi-discrete equation (10) with respect to t by the backward Euler method to obtain a full discrete weak Galerkin finite element method:seek Un∈ Vh0(n= 0,1,2, …, N) such that U0= Qhψ and where (?)Un= (Un-Un-1)/k. This is equivalent toThen, we study the stability, energy conservation and convergence of the numerical algorithms (13) and (14) of the parabolic problem (8), and present numerical examples. Our main results are as follows:Theorem 7. Let uh(t)={u0(·,t),ub(·,t)} be the solution of the semi-discrete weak Galerkin finite element method (13) for the parabolic problem. Then there exists a constant C> 0 independent of the mesh size h such that i.e. the numerical solution is stable with respect to initial approximate value and forcing term.Theorem 8. Let uh(t)={u0(·, t), ub(·, t)} be the solution of the semi-discrete weak Galerkin finite element method (13) for the parabolic problem. Then uh(t) has the following energy preserving property on each T E Th, i.e., For every T∈Th where and denote by Rh the L2 projection of [L2(T)]2 onto Gr-1(T).Theorem 9. Let u ∈ Hr+1(Ω) be the solution of the parabolic problem (8) and uh be the solution of the semi-discrete weak Galerkin finite element method (13) for the parabolic problem. Denote by e:= uh - Qhu ∈ Vh0 the difference between the weak Galerkin approximation uh and the L2 projection of the exact solution u. Assume u∈Hr+1(Ω). Then there exists a constant C> 0 independent of the mesh size h such that the following estimates hold andTheorem 10. Assume that the finite element partition Th is shape regular. Then there exists a constant C> 0 independent of the mesh size h such thatTheorem 11. Let u∈Hr+1(Ω) be the solution of the parabolic problem (8) and Un be the solution of full discrete weak Galerkin finite element method for the parabolic problem (14), respectively. Denote by en:= Un - Qhu(tn) the difference between the backward Euler weak Galerkin approximation and the 1? projection of the exact solution u. Assume u∈C2([0, t0]; Hr+1(Ω)). Then there exists a constant C> 0 independent of the mesh size h such that for 0<n≤ N the following inequalities hold and whereTheorem 12. Let u ∈ H1+r(Ω) and the corresponding elliptic problem of parabolic problem (8) has the H2-regularity. Then there exists a constant C> 0 independent of the mesh size h such that andTheorem 13. u∈Hr+1(Ω). Then there exists a constant C> 0 independent of the mesh size h such that...