Conservative Local Discontinuous Galerkin Method for Compressible Miscible Displacements in Porous Media

Abstract

In Guo et al. (Appl Math Comput 259:88–105, 2015), a nonconservative local discontinuous Galerkin (LDG) method for both flow and transport equations was introduced for the one-dimensional coupled system of compressible miscible displacement problem. In this paper, we will continue our effort and develop a conservative LDG method for the problem in two space dimensions. Optimal error estimates in $L^{\infty }(0, T; L^{2})$ norm for not only the solution itself but also the auxiliary variables will be derived. The main difficulty is how to treat the inter-element discontinuities of two independent solution variables (one from the flow equation and the other from the transport equation) at cell interfaces. Numerical experiments will be given to confirm the accuracy and efficiency of the scheme.

Appendix: Proof of Lemma 3.1

Recall that we have chosen the initial condition $c_{h}^{0}=P^{+}c_{0}, \mathbf {u}_{h}^{0}=\mathbf {\Pi ^{-}}\mathbf {u}_{0}$, where $\mathbf {u}_{0}=-a(c_{0})\nabla p_0$, and $\widehat{p_{h}}=p_{h}^{+}, \widehat{\mathbf {u}_{h}}=\mathbf {u}_{h}^{-}, \widehat{\mathbf {z}_{h}}=\mathbf {z}_{h}^{-}, \widehat{c_{h}}=c_{h}^{+}$. For simplicity, we will drop the 0 in the superscripts and subscripts in this section. It is clear that (3.15) and (3.16) hold. Taking the test function $\zeta ={\xi _{p}}_{t}$ and summing over K in (4.10), we have

$$\begin{aligned}&\Big (d(c){\xi _{p}}_{t},{\xi _{p}}_{t}\Big )=\Big (d(c){\eta _{p}}_{t},{\xi _{p}}_{t}\Big )+\Big ({p_{h}}_t(d(c)-d(c_h)),{\xi _{p}}_{t})\Big ), \end{aligned}$$

(A.1)

where we have used $\mathbf {u}_{h}=\mathbf {\Pi ^{-}}\mathbf {u}, \widehat{\mathbf {u}_{h}}=\mathbf {u}_{h}^{-}$ and the property of the projection (4.3). Using the Schwartz inequality, we can get

$$\begin{aligned} \Vert d^{\frac{1}{2}}(c){\xi _{p}}_{t}\Vert ^{2}\le C\Vert {\eta _{p}}_{t}\Vert \Vert {\xi _{p}}_{t}\Vert +C\Vert c-c_{h}\Vert \Vert {\xi _{p}}_{t}\Vert , \end{aligned}$$

(A.2)

By Lemma 4.2 and (3.15), we easily prove

$$\begin{aligned} \Vert {\xi _{p}}_{t}\Vert \le Ch^{k+1}. \end{aligned}$$

(A.3)

Similarly, taking the test function $\mathbf {w}=\varvec{\xi }_{s}$ and summing over K in (4.7), we have

$$\begin{aligned} (\varvec{\xi }_{s},\varvec{\xi }_{s})=(\varvec{\eta }_{s},\varvec{\xi }_{s})-\mathcal {D}(\eta _{c},\varvec{\xi }_{s}), \end{aligned}$$

(A.4)

where we have used $c_{h}=P^{+}c$. Using the Schwartz inequality and the Lemma 4.3, we can get

$$\begin{aligned} \Vert \varvec{\xi }_{s}\Vert ^{2}\le \Vert \varvec{\xi }_{s}\Vert \Vert \varvec{\eta }_{s}\Vert +Ch^{k+1}\Vert c\Vert _{k+2}\Vert \varvec{\xi }_{s}\Vert . \end{aligned}$$

(A.5)

By Lemma 4.2, we easily prove

$$\begin{aligned} \Vert \varvec{\xi }_{s}\Vert \le Ch^{k+1}, \end{aligned}$$

(A.6)

By the standard approximation results, (3.17) and (3.18) hold. At last we estimate $p-p_{h}$, following the technique in [17]. By (3.9) the initial data $p_{h}$ is the solution of the following equations

$$\begin{aligned}&(A(c_{h})\mathbf {u}_{h},\varvec{\theta })_K-(p_{h},\nabla \cdot \varvec{\theta })_K +\langle \widehat{p_{h}},\varvec{\theta }\cdot \varvec{\nu }_{K}\rangle _{\partial _K}=0, \end{aligned}$$

(A.7)

and also satisfies

$$\begin{aligned}&(p-p_{h},1)=0. \end{aligned}$$

(A.8)

From (4.9), we have

$$\begin{aligned}&(A(c)\mathbf {u}-A(c_{h})\mathbf {u}_{h},\varvec{\theta })_K-(p-p_{h},\nabla \cdot \varvec{\theta })_K +\langle p-\widehat{p_{h}},\varvec{\theta }\cdot \varvec{\nu }_{K}\rangle _{\partial _K}=0. \end{aligned}$$

(A.9)

We use $\mathbf {u}_{h}$ to find a well-defined $p_{h}$, and we only need to prove the uniqueness. If there are two solutions $p_{1}$ and $p_{2}$ satisfying (A.7) and (A.8), then we can easily get

$$\begin{aligned}&(p_{1}-p_{2},\nabla \cdot \varvec{\theta })_K -\langle \widehat{p_{1}}-\widehat{p_{2}},\varvec{\theta }\cdot \varvec{\nu }_{K}\rangle _{\partial _K}=0, \end{aligned}$$

(A.10)

$$\begin{aligned}&(p_{1}-p_{2},1)=0. \end{aligned}$$

(A.11)

We consider the elliptic linear problem

$$\begin{aligned}&-\varvec{\zeta }^{*}=\nabla {\xi }^{*}, \quad \; \text{ in }\quad \Omega , \end{aligned}$$

(A.12)

$$\begin{aligned}&{\eta }^{*}=\nabla \cdot \varvec{\zeta }^{*}, \quad \; \text{ in }\quad \Omega , \end{aligned}$$

(A.13)

subject to periodic boundary conditions. To make the problem well-defined, we assume that the average of ${\xi }^{*}$ on $\Omega$ is a given constant and that of $\eta ^{*}$ is zero. We have the elliptic regularity result

$$\begin{aligned} \Vert \varvec{\zeta }^{*}\Vert _{H^{1}(\Omega )}+\Vert {\xi }^{*}\Vert _{H^{2}(\Omega )}\le C\Vert {\eta }^{*}\Vert . \end{aligned}$$

(A.14)

Taking $\eta ^{*}=p_{1}-p_{2}$ and $\widehat{p_i}=p_i^{+}, i=1,2$, we get

$$\begin{aligned}&(p_{1}-p_{2},p_{1}-p_{2})_{K}\nonumber \\&\quad =(p_{1}-p_{2},\nabla \cdot \varvec{\zeta }^{*})_K\nonumber \\&\quad = (p_{1}-p_{2},\nabla \cdot (\varvec{\zeta }^{*}-\Pi \varvec{\zeta }^{*}))_K+(p_{1}-p_{2},\nabla \cdot \Pi \varvec{\zeta }^{*})_K\nonumber \\&\quad = (p_{1}-p_{2},\nabla \cdot (\varvec{\zeta }^{*}-\Pi \varvec{\zeta }^{*}))_K-\langle \widehat{p_{1}}-\widehat{p_{2}},(\varvec{\zeta }^{*}-\Pi \varvec{\zeta }^{*}) \cdot \varvec{\nu }_{K}\rangle _{\partial _K} +\langle \widehat{p_{1}}-\widehat{p_{2}},\varvec{\zeta }^{*} \cdot \varvec{\nu }_{K}\rangle _{\partial _K}\nonumber \\&\quad = -(\nabla (p_{1}-p_{2}),\varvec{\zeta }^{*}-\Pi \varvec{\zeta }^{*})_K+\langle p_{1}-p_{2},(\varvec{\zeta }^{*}-\Pi \varvec{\zeta }^{*})\cdot \varvec{\nu _K}\rangle _{\partial _K}\nonumber \\&\qquad -\langle \widehat{p_{1}}-\widehat{p_{2}},(\varvec{\zeta }^{*}-\Pi \varvec{\zeta }^{*}) \cdot \varvec{\nu }_{K}\rangle _{\partial _K} +\langle \widehat{p_{1}}-\widehat{p_{2}},\varvec{\zeta }^{*} \cdot \varvec{\nu }_{K}\rangle _{\partial _K}\end{aligned}$$

(A.15)

where the third step follows from (A.10) and the last equality is based on integration by parts. We take $\Pi \varvec{\zeta }^{*}=\Pi ^{-}\varvec{\zeta }^{*}$ and sum over K. By the continuity of $\varvec{\zeta }^{*}$ and the definition of the projection $\Pi ^{-}$, we obtain

$$\begin{aligned} (p_{1}-p_{2},p_{1}-p_{2})=0 \end{aligned}$$

(A.16)

Then we get $p_{1}=p_{2}$. We have proved that $p_{h}$ is well-defined. In the following, we estimate $\Vert p-p_{h}\Vert$. We use the same technique above and take $\eta ^{*}=p-p_{h}$ to obtain

$$\begin{aligned}&(p-p_{h},p-p_{h})_{K}\nonumber \\&\quad = (p-p_{h},\nabla \cdot \varvec{\zeta }^{*})_K\nonumber \\&\quad = (p-p_{h},\nabla \cdot (\varvec{\zeta }^{*}-\Pi \varvec{\zeta }^{*}))_K+(p-p_{h},\nabla \cdot \Pi \varvec{\zeta }^{*})_K\nonumber \\&\quad =(p-p_{h},\nabla \cdot (\varvec{\zeta }^{*}-\Pi \varvec{\zeta }^{*}))_K -(A(c)\mathbf {u}-A(c_h)\mathbf {u}_h,\varvec{\zeta }^{*}-\Pi \varvec{\zeta }^{*})_K\nonumber \\&\qquad -\langle p-\widehat{p_h},(\varvec{\zeta }^{*}-\Pi \varvec{\zeta }^{*})\cdot \varvec{\nu _K}\rangle _{\partial K} +(A(c)\mathbf {u}-A(c_h)\mathbf {u}_h,\varvec{\zeta }^{*})_K +\langle p-\widehat{p_h},\varvec{\zeta }^{*}\cdot \varvec{\nu _K}\rangle _{\partial K} \nonumber \\&\quad =-(\nabla (p-p_{h}),\varvec{\zeta }^{*}-\Pi \varvec{\zeta }^{*})_K +\langle p-p_h,(\varvec{\zeta }^{*}-\Pi \varvec{\zeta }^{*})\cdot \varvec{\nu _K}\rangle _{\partial K} \nonumber \\&\qquad -\,(A(c)\mathbf {u}-A(c_h)\mathbf {u}_h,\varvec{\zeta }^{*}-\Pi \varvec{\zeta }^{*})_K -\langle p-\widehat{p_h},(\varvec{\zeta }^{*}-\Pi \varvec{\zeta }^{*})\cdot \varvec{\nu _K}\rangle _{\partial K}\nonumber \\&\qquad +\,(A(c)\mathbf {u}-A(c_h)\mathbf {u}_h,\varvec{\zeta }^{*})_K +\langle p-\widehat{p_h},\varvec{\zeta }^{*}\cdot \varvec{\nu _K}\rangle _{\partial K}\nonumber \\&\quad =-(\nabla (p-p_{h}),\varvec{\zeta }^{*}-\Pi \varvec{\zeta }^{*})_K +\langle \widehat{p_h}-p_h,(\varvec{\zeta }^{*}-\Pi \varvec{\zeta }^{*})\cdot \varvec{\nu _K}\rangle _{\partial K} \nonumber \\&\qquad -\,(A(c)\mathbf {u}-A(c_h)\mathbf {u}_h,\varvec{\zeta }^{*}-\Pi \varvec{\zeta }^{*})_K +(A(c)\mathbf {u}-A(c_h)\mathbf {u}_h,\varvec{\zeta }^{*})_K \nonumber \\&\qquad +\, \langle p-\widehat{p_h},\varvec{\zeta }^{*}\cdot \varvec{\nu _K}\rangle _{\partial K} \end{aligned}$$

(A.17)

where the third one follows from (A.9) and the fourth equality is based on the integrate by parts. Recalling that $\widehat{p_{h}}=p_{h}^{+}$, we take $\Pi \varvec{\zeta }^{*}=\Pi ^{-}\varvec{\zeta }^{*}$ and sum over K. By the continuity of $\varvec{\zeta }^{*}$ and the definition of the projection $\varvec{\Pi }^{-}$, we obtain

$$\begin{aligned} \Vert p-p_{h}\Vert ^{2}= & {} -(\nabla \eta _p^{},\varvec{\zeta }^{*}-\Pi \varvec{\zeta }^{*})-(A(c)\mathbf {u}-A(c_h)\mathbf {u}_h,\varvec{\zeta }^{*}-\Pi \varvec{\zeta }^{*})\nonumber \\&+(A(c)\mathbf {u}-A(c_h)\mathbf {u}_h,\varvec{\zeta }^{*})\nonumber \\= & {} -(\nabla {\eta _p},\varvec{\zeta }^{*}-\Pi \varvec{\zeta }^{*})-(A(c)(\mathbf {u}-\mathbf {u}_h),\varvec{\zeta }^{*}-\Pi \varvec{\zeta }^{*})\nonumber \\&-((A(c)-A(c_h))\mathbf {u}_h,\varvec{\zeta }^{*}-\Pi \varvec{\zeta }^{*})\nonumber \\&+(A(c_0)(\mathbf {u}-\mathbf {u}_h),\varvec{\zeta }^{*})+((A(c)-A(c_h))\mathbf {u}_h,\varvec{\zeta }^{*})\nonumber \\\le & {} Ch^{k+1}\Vert \varvec{\zeta }^{*}\Vert _{H^{1}(\Omega )}+Ch^{k+2}\Vert \varvec{\zeta }^{*}\Vert _{H^{1}(\Omega )}+Ch^{k+1}\Vert \varvec{\zeta }^{*}\Vert \nonumber \\\le & {} Ch^{k+1}\Vert \varvec{\zeta }^{*}\Vert _{H^{1}(\Omega )}\nonumber \\\le & {} Ch^{k+1}\Vert p-p_{h}\Vert , \end{aligned}$$

(A.18)

which further implies

$$\begin{aligned} \Vert p-p_{h}\Vert \le Ch^{k+1}. \end{aligned}$$

(A.19)