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
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