1. Introduction

In this paper we consider the following convection dominated Sobolev equation:

where Ω is a bounded convex domain in ℝm with 1 ≤ m ≤ 3 with boundary ∂Ω, c(x), d(x), a(x), b(x), f(x, t), and u0(x) are given functions. The Sobolev equation which represents the flow of fluids through fissured rock, the migration of the moisture in soil, the physical phenomena of thermodynamics and other applications as described in [2,19,20], is one of most principal partial differential equations. For the existence and uniqueness results of the solutions of the equation (1.1), refer to [8].

For the problems with no convection term, mixed finite element methods [11,16,18,22], least-squares methods [12,18,21,22], and discontinuous Galerkin methods [14,15] were used for numerical treatments. In the case that a conventional (least-squares) MFEM is applied, we generally needs to solve the coupled system of equations in two unknowns, which brings to difficulties in some extent. So, in [18], a split least-squares mixed finite element method for reaction-diffusion problems was firstly introduced to solve the uncoupled systems of equations in the unknowns.

For the partial differential equations with a convection term, a characteristic (mixed) finite element method is one of the useful methods [1,3,4,5,6,7,10,13] because it reflects well the physical character of a convection term and also it treats efficiently both convection term and time derivative term. Gao and Rui [9] introduced a split least-squares characteristic MFEM to approximate the primal unknown u and the flux unknown −a∇u of the equation (1.1) and obtained the optimal convergence in L2(Ω) norm for the primal unknown and in H(div, Ω) norm for the flux unknown. And Zhang and Guo [23] introduced a split least-squares characteristic mixed element method for nonlinear nonstationary convection-diffusion problem to approximate the primal unknown and the flux unknown and obtained the optimal convergence in L2(Ω) norm for the primal unknown and in H(div, Ω) norm for the flux unknown.

In this paper, we apply a split least-squares characteristic characteristic mixed finite element method (MFEM) to achieve two uncoupled system of equations, one of which is for approximations to the primal unknown u and the other of which is for ones to the flux unknown σ = −(a(x)∇ut+b(x)∇u) of the equation (1.1). And we analyze the optimal order of convergence in L2 and H1 normed spaces for the approximations. In section 2, we introduce necessary assumptions and notations, and in section 3, we construct finite element spaces on which we compose the approximations of two unknowns. In section 4, by adopting a split least-squares characteristic MFEM, we construct the approximations of the primal unknown and the unknown flux and establish the convergence of optimal order in L2 and H1 normed spaces for the primal unknown and the convergence of optimal order in L2 normed space for the flux unknown. In section 5, we provide some numerical results to confirm the validity of the theoretical results obtained in section 4.

 

2. Assumption and notations

For an s ≥ 0 and 1 ≤ p ≤ ∞, we denoted by Ws,p(Ω) the Sobolev space endowed with the norm where k = (k1, k2, · · · , km), |k| = k1 +k2 +· · ·+km, and ki is a nonnegative integer, for each i, 1 ≤ i ≤ m. If p = 2, we simply denote Hs(Ω) = Ws,2(Ω) and ║ϕ║s = ║ϕ║s,2. And also in case that s = 0, we simply write ║ϕ║. We let Hs(Ω) = {u = (u1, u2, · · · , um) | ui ∈ Hs(Ω), 1 ≤ i ≤ m} with the norm and W = H(div, Ω).

If ϕ(x, t) belongs to a Sobolev space equipped with a norm ║·║X for each t, then we let

In case that t0 = T, we denote Lp(0, T : X) and L∞(0, T : X) by Lp(X) and L∞(X), respectively. Let Hq,∞(X) = {ϕ(x, t) | ϕ(x, t), ϕt(x, t), · · · , ϕq(x, t) ∈ L∞(X)} for a nonnegative integer q.

We consider the problem (1.1) with the coefficients satisfying the following assumption:

(A). There exist c∗, c∗, d∗, a∗, a∗, b∗, and b∗ such that 0 < c∗ < c(x) ≤ c∗, 0 < |d(x)| ≤ d∗, 0 < a∗ < a(x) ≤ a∗, and 0 < b∗ < b(x) ≤ b∗, for all x ∈ Ω, where

 

3. Finite element spaces

Before preceding the numerical scheme, we let ℇh = {E1,E2, · · · ,ENh} be a family of regular finite element subdivision of Ω. We let h denote the maximum of the diameters of the elements of ℇh. If m = 2, then Ei is a triangle or a quadrilateral, and if m = 3, then Ei is a 3-simplex or 3-rectangle. Boundary elements are allowed to have a curvilinear edge (or a curved surface).

We denote by Vh × Wh the Raviart-Thomas-Nedlec space associated with ℇh. For each triangle (or 3-simplex) element E ∈ ℇh, we define Vh(E) = Pk(E), and Wh(E) = Pk(E)m ⊕ (x1, x2, · · · , xm)T Pk(E) where Pk(E) is the set of polynomials of total defree ≤ k difined on E. Now we define the finite element spaces

And also in case that E is a rectangle (or a parallelogram), we adopt analogous modification to construct Vh and Wh.

Let Ph×∏h : V × W → Vh×Wh denote the Raviart-Thomas [17] projection which satisfies

Then, obviously, (∇ · w, v − Phv) = 0 holds for each v ∈ V and each w ∈ Wh and div∏h = Phdiv is a function from W onto Vh. It is proved that the following approximation properties hold [17]:

 

4. Optimal L2 error analysis

Let and ν = ν(x, t) be the unit vector in the direction of (d(x), c(x)). Then, we have

Hence the problem (1.1) can be written in the form

By introducing the flux term σ = −(a(x)∇ut +b(x)∇u), the problem (4.1) can be rewritten as follows:

For a positive integer N, let Δt = T/N and tn = nΔt, n = 0, 1, · · · ,N. Choosing t = tn in (4.2) and discretizing it with respect to t by applying the backward Euler method along ν-characteristic tangent at (x, tn), we get

where . Therefore we have

where

Now let ã(x) = a(x) + b(x)△t. By multiplying the first equation of (4.3) by and the second equation by , we have the equivalent system of equations

For (v, τ) ∈ V × W, we define a least-squares functional J(v, τ) as follows

Then the least-squares minimization problem is to find s solution (un,σn) ∈ V × W such that

If we define the bilinear form A on (V × W)2 by

then the weak formulation of the minimization problem becomes as follows: find (un,σn) ∈ V × W such that

Based on (4.6), we derive the following least-squares characteristic MFEM scheme: find ∈ Vh × Wh satisfying

Lemma 4.1. For (v, τ) ∈ V × W, we have

Proof. From the definition of the bilinear form (4.5), we have

Letting vh = 0 in (4.7) and applying the definition of the bilinear form A, we have

which implies that

Since , we have

Letting τh = 0 in (4.7) and applying the definition of the bilinear form A, we have

Finally, we derive a split least-squares characteristic MFEM: find ∈ Vh × Wh satisfying:

For the error analysis, we define an elliptic projection ũ(x, t) of u(x, t) onto Vh satisfying

Obviously, by the assumption (A), there exists a unique elliptic projection ũ(x, t) ∈ Vh. Now we let η = u − ũ and ξ = uh − ũ so that u − uh = η − ξ.

Hereafter a constant K denotes a generic positive constant depending on Ω and u, but independent of h and Δt, and also any two Ks in different places don’t need to be the same. We state the error bounds of η below, the proofs of which can be found in [14,15].

Theorem 4.2 ([14]). If ut ∈ L2(Hs(Ω)) and u0 ∈ Hs(Ω), then there exists a constant K, independent of h, such that

(i) ║η║ + h║η║1 ≤ Khμ(║ut║L2(Hs) + ║u0║s),

(ii) ║η║ + h║ηt║1 ≤ Khμ(║ut║L2(Hs) + ║u0║s),

where μ = min(k + 1, s).

Theorem 4.3 ([15]). If ut ∈ L2(Hs(Ω)), utt(t) ∈ Hs(Ω), and u0 ∈ Hs(Ω), then there exists a constant K, independent of h, such that

where μ = min(k + 1, s).

Lemma 4.4. If u ∈ H1,∞(H2(Ω)) and utt(t) ∈ L2(Ω), then

Proof. By applying Taylor’s expansion, we obviously have the estimations of . □

Theorem 4.5. In addition to the hypotheses of Theorem 4.2 and 4.3, if u(t) ∈ Hs(Ω), u ∈ H1,∞(H2(Ω)), and Δt = O(h), then

where μ = min(k + 1, s).

Proof. Subtracting (4.1) at t = tn from (4.8), we get the equation

Now we set in (4.11). Then letting three terms of the left-hand side of (4.11) by L1,L2, and L3, respectively, we get the following estimates for L1,L2, and L3

Now let ϵ > 0 be sufficiently small, but independent of h and Δt. Since

for some , R1 can be estimated as follows:

By noting that

for some , we can estimate R2 as follows:

By Lemma 4.4, we obviously get

By (4.10), Theorem 4.3, and the Taylor expansion, we have

where . By the Taylor expansion, we get

for some . Now by applying the bounds of L1 ∼ L3 and R1 ∼ R6 to (4.11), we obtain

which yields that for sufficiently small ϵ > 0

Now we sum up both sides of (4.12) from n = 1 to n = N to get

By the discrete-type Gronwall inequality, we get

from which we get by Poincare’s inequality

Therefore, by using Theorem 4.2 and the triangular inequality, we obtain

By applying Lemma 4.1 to (4.6), we get

and hence, letting v = 0, we obtain

And letting vh = 0 in (4.7) and applying the definition of the bilinear form A, we get

which implies that

Therefore we have

For σ ∈ W, we define an elliptic projection ∈Wh of σ satisfying

where λ is a positive real number. By applying the Lax-Milgram lemma, the existence of can be obtained.

Lemma 4.6. If σ ∈ W ∩ Hs(Ω), then there exists a constant K > 0 such that

where μ = min(k + 1, s).

Proof. By the difinition of and (3.5), we get

and so

Therefore, by (3.4), we have

for sufficiently small λ > 0. We let φ ∈ H2(Ω) be the solution of an elliptic problem

where n denotes the outward normal unit vector to ∂Ω. By the regularity property of the elliptic problem, we have ║φ║2 ≤ K║σ − ∥. Using (3.4), (3.5), (4.15), (4.16), and (4.17), we obtain the following estimation

Now if we choose h sufficiently small, then we get ║σ − ║ ≤ Khμ║σ║s. □

Theorem 4.7. In addition to the hypotheses of Theorem 4.5, if σ ∈ W ∩Hs(Ω), then

where μ = min(k + 1, s).

Proof. By subtracting (4.14) from (4.13), we have

Now we let π = σ − , ρ = − σh. From (4.18), we get

Choosing τh = ρn in (4.19) and applying the integration by parts, we obtain

Note that

and

By applying (4.15) to (4.20), we get

By using Lemma 4.4, Lemma 4.6, and Theorem 4.5, we get

Let ψn ∈ H2(Ω) be the solution of an elliptic problem

where n denotes the outward normal unit vector to ∂Ω. By the regularity property of the elliptic problem, we have ║ψn║2 ≤ K║ρn║. We let be the elliptic projection of ψn onto Wh defined by exactly the same way as (4.15). Then using (4.19) and (4.22) with τh = , we get

By using (4.21), Lemma 4.6, and the fact that ║ψn − ║ ≤ ch2║ψn║2, we get the estimations of I1 ∼ I3 as follows:

By the definitions of and Theorem 4.5, we have the estimations of I4 ∼ I6 as follows:

Using the definition ψn, Theorem 4.2, and Lemma 4.4, we estimate I7 ∼ I9 as follows:

By applying the estimations of I1 ∼ I9 to (4.23), we obtain

Therefore ║ρn║ ≤ K(hμ + Δt) holds for sufficiently small h > 0. Thus by the triangular inequality and Lemma 4.6, we obtain the result of this theorem. □

 

5. Numerical example

In this section, we will present some numerical results to verify the convergence order of the split least-squares CMFEM proposed in (4.8) and (4.9). For the sake of convenience, we consider the one dimensional convection dominated Sobolev equation (1.1) with c(x) = d(x) = 1, a(x) = b(x) = 0.001 and Ω = [0, 1].

We construct the approximation of u(x, t) on the finite element space consisting of the piecewise linear polynomials defined on the uniform grids and the approximation of σ(x, t) on the finite element space consisting of the piecewise quadratic polynomials defined on the uniform grids. Choose the exact solution u(x, t) as follows:

and compute f(x, t) = ut + ux − 10−3uxx − 10−3utxx by substituting u(x, t) defined in (5.1). Notice that u(x, t) ∈ H4(Ω) and σ(x, t) ∈ H2(Ω)

The numerical results for uh(x) at T = 0.4 are given in Table 1 in terms of the space mesh size h and the time mesh size △t. We know from Table 1 that the convergence orders in L2 and H1 norms for uh at T = 0.4 are consistent with the results in Theorem 4.5.

Table 1.The estimates for uh

The corresponding numerical results for σh at T = 0.4 are given in Table 2 in terms of the space mesh size h and the time mesh size △t. We know from Table 2 that the convergence order in L2 norm for σh at T = 0.4 is consistent with the result in Theorem 4.7.

Table 2.The estimates for σh