Abstract
This paper aims to investigate the unconditionally optimal and superconvergent error estimates of a mass- and energy-conserved finite element method for the Schrödinger–Poisson equation. Firstly, a priori error bound of the numerical solutions in \(H^1\)-norm is obtained by the conserved property. Secondly, the unconditionally optimal error estimates in \(L^2\)-norm are derived without any timestep restriction in terms of the bound of the numerical solution. Thirdly, the unconditionally superclose error estimates in \(H^1\)-norm are got by treating the coupled nonlinear term rigorously and skillfully. Furthermore, the unconditionally superconvergent error estimates in \(H^1\)-norm are acquired by the interpolation post-processing approach. Finally, some numerical results are provided to verify the theoretical analysis.
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1 Introduction
In this paper, we consider the following two dimensional Schrödinger–Poisson (SP) equation:
where \(u=u(\varvec{x},t)\) is a complex-valued function with respect to time t and spatial variable \(\varvec{x}=(x,y)\in \Omega \), which is a bounded rectangular domain in \(\mathbb {R}^2\), \(\mu =\pm 1\) is a rescaled physical constant, which signifies the property of the underlying forcing, repulsive if \(\mu >0\) and attractive if \(\mu <0\) (Yi and Liu 2022). \(\textrm{i}=\sqrt{-1}\) denotes the imaginary unit and \(T>0\) is the final time.
The SP equation can be employed in many physical applications, including semiconductors (Ringhofer and Soler 2000; Markowich et al. 1990), plasma physics (Shukla and Eliasson 2011) and cosmology (Uhlemann et al. 2014). System (1.1)–(1.4) preserves both the mass and the energy. It is an important and interesting thing to design numerical schemes that satisfy discrete analogues of these laws, as typically this leads to good qualitative behaviour of numerical solutions for longer computational times (Athanassoulisa et al. 2023). There exists a very large literature on numerical methods and analysis for the SP equation. A conservative discontinuous Galerkin scheme was developed in Yi and Liu (2022) for the SP equation and the corresponding optimal \(L^2\) error estimates were obtained. With the help of a Crank–Nicolson temporal and finite difference spatial discretization, a predictor–corrector scheme was studied in Ringhofer and Soler (2000). In Auzinger et al. (2017), a rigorous stability and error analysis was presented in terms of the second-order Strang splitting finite element discretization. The convergence rates were established for the periodic SP equation based on a Galerkin approximation in Bohun et al. (1996). An error analysis of Strang-type splitting integrators was discussed in detail for Schrödinger–Poisson and cubic nonlinear Schrödinger equations in Lubich (2008). Moreover, a second order convergence of the Strang splitting method was discussed in Auzinger et al. (2017) for Schrödinger–Poisson equation.
The objective of this work is to develop a structure-preserving fully-discrete Galerkin scheme for the SP equation, which preserves both mass and energy at the discrete level. In particular, for the spatial discretization, we adopt the standard conforming finite element method, while for the temporal discretization, we use the Crank–Nicolson method. The main advantage of the proposed scheme is that it avoids the grid ratio restrictions between temporal step size and spatial step size, while some certain restriction required in the previous literature. More precisely, a priori error bound in \(H^1\)-norm rather than the \(L^{\infty }\)-norm is derived according to the mass- and energy conserved properties. Then, by treating the nonlinear and coupled term rigorously and skillfully, the unconditionally optimal error estimates in \(L^2\)-norm and the superconvergent error estimates in \(H^1\)-norm are established.
The rest of this paper is organized as follows. In Sect. 2, we introduce some preliminaries and lemmas, which are needed in the error analysis. In Sect. 3, the unconditionally optimal error estimates in \(L^2\)-norm are presented for the conserved Crank–Nicolson fully-discrete finite element scheme. In Sect. 4, the unconditionally superconvergent error estimates in \(L^2\)-norm are studied. In Sect. 5, some numerical experiments are carried out to confirm the theoretical analysis.
2 Some preliminaries and lemmas
Let \(W^{m,p}(\Omega )\) be the standard Sobolev space (Adams and Fournier 2003) with the norm \(\Vert \cdot \Vert _{m,p}\) and semi-norm\(|\cdot |_{m,p}\). For any two complex functions \(u,~v\in L^2(\Omega )\), we define the \(L^2(\Omega )\) inner product by \( (u,v)=\int _{\Omega }u(\varvec{x})(v(\varvec{x}))^*d\varvec{x}, \) where \(v^{*}\) denotes the conjugate of v. Moreover, for any Banach space Y and function \(f: [0,T]\rightarrow Y\), define the norm
Let \(\mathcal {T}_h\) be a uniform rectangular partition of \(\Omega \) into rectangles \(\{K\}\) and \(h=\max _{K\in \mathcal {T}_h}\{\text{ diam } (K)\}\) be the mesh size. For a given element \(K\in \mathcal {T}_h\), we define the bilinear finite element space
Moreover, define \(R_h: H_0^1(\Omega )\rightarrow V_h\) to be the Ritz projection operator by
Then, by the classical finite element theory (Thomee 2006; Brenner and Scott 2002), there holds for \(u\in H^2(\Omega )\cap H_0^1(\Omega )\) that
The weak formulation of the problem (1.1)–(1.4) reads: find \(u:~[0,T]\rightarrow H_0^1(\Omega )\) and \(\Phi :~[0,T]\rightarrow H_0^1(\Omega )\), such that
In order to present the fully-discrete scheme, let \(\{t_n|~t_n=n\tau ;0\le n \le N\}\) be a uniform partition in time with time step \(\tau =T/N\) and \(f^n=f(\varvec{x},t_n)\). For a sequence of functions \(\{f^n\}_{n=0}^N\), we denote
Then, the fully-discrete scheme is: for given \(u_h^{n-1}\in V_h\) and \(\Phi _h^{n-1}\in V_h\), find \(u_h^{n}\in V_h\) and \(\Phi _h^{n}\in V_h\), such that
with the initial approximations \(u_h^0\) and \(\Phi _h^0\) defined by
Lemma 1
The numerical scheme (2.5)–(2.6) has the following mass and energy-conversed properties
where
Proof
Choosing \(v_h=\bar{u}_h^n\) in (2.5) and taking the imaginary parts of the resulting equation give that
which shows that
Clearly, by the definition of \(\mathcal {M}^n\), the mass conservation is obtained. Moreover, choosing \(v_h=D_{\tau }u_h^n\) in (2.5) and taking the real parts of the resulting equation result in
Note that
one can get
Substituting (2.11) into (2.10) yields that
On the other hand, from (2.6) at \(t=t_n\) and \(t=t_{n-1}\), we have
Then, choosing \(w_h=\bar{\Phi }_h^n\) in (2.13) leads to
Substituting (2.14) into (2.12) gives that
Then, by the definition of \(\mathcal {E}^n\), we obtain the energy conservation. The proof is complete.
Lemma 2
Suppose that \(u_0\in H_0^1(\Omega )\), we have the following a priori error bound
where C is a constant independent of n, h and \(\tau \).
Proof
From Lemma 1, one can check that
Choosing \(w_h=\Phi _h^0\) in (2.6) at \(t=t_0\) yields that
Thus, we have
Moreover, choosing \(w_h=\Phi _h^n\) in (2.6) at \(t=t_n\) gives that
where we have used (2.9), Sobolev inequality \(\Vert \chi \Vert _{0,4}^2\le C\Vert \chi \Vert _0\Vert \nabla \chi \Vert _0\), for \(\chi \in H_0^1(\Omega )\), \(H^1(\Omega )\hookrightarrow L^4(\Omega )\) and Poincare inequality in the above estimate. From (2.18), it is not difficult to see that
where we have used (2.9) again in the above estimate.
Hence, by (2.16), (2.17) and (2.19), we have
Hence, the desired result (2.15) is obtained by Poincare inequality.
Next, we present the discrete Gronwall inequality, which is an important tool for analyzing time-dependent problems.
Lemma 3
(Gronwall’s inequality Heywood and Rannacher 1990; Riviére 2008) Let \(\tau \), B, \(C>0\) and let \(\{a_n\}\), \(\{b_n\}\), \(\{c_n\}\) be sequences of nonnegative numbers satisfying
Then, if \(C\tau <1\), there holds
Remark 1
Note that \((n+1)\tau \le 2T\), one can see that the constant in the above Gronwall’s inequality is exponentially dependent on the final time T.
3 Unconditionally optimal error estimate in \(L^2\)-norm of the fully-discrete scheme
We present the first main result in the following theorem.
Theorem 3.1
Suppose that \((u^n,\Phi ^n)\) and \((u_h^n,\Phi _h^n)\) are the solutions of (2.3)–(2.4) and (2.5)–(2.6) at \(t=t_n\), respectively. Moreover, suppose that \(u, u_t, u_{tt}\in L^{\infty }(H^2(\Omega ))\), \(u_{ttt}\in L^{\infty }(L^2(\Omega ))\), \(\Phi \in L^{\infty }(H^2(\Omega ))\), \(\Phi _{tt}\in L^{\infty }(L^2(\Omega ))\). Then we have the following unconditionally optimal error estimate
Proof
For the sake of simplicity, we split the errors \(u^n-u_h^n\) and \(\Phi ^n-\Phi _h^n\) as:
From (2.3)–(2.4) and (2.5)–(2.6), we have the following error equations:
Choosing \(v_h=\bar{\eta }^n\) in (3.2) and taking the imaginary parts result in
where we have used the definition of Ritz projection.
By the Cauchy–Schwarz inequality and (2.2), \(A_1\) can be bounded by
In order to estimate \(A_2\), we rewrite \(\bar{\Phi }^n\bar{u}^n-\bar{\Phi }_h^n\bar{u}_h^n\) as
One can easily see that
By Hölder inequality, we have
where we have used Lemma 2 and the Sobolev inequality. Similarly, we have
Based on the estimates (3.7)–(3.9), \(A_2\) can be bounded by
According to Taylor expansion and integration by parts, we have
Substituting (3.5), (3.10) and (3.11) into (3.4) yields that
On the other hand, choosing \(w_h=\theta ^n\) in (3.3) leads to
where we have used the definition of Ritz projection. Note that
one can check that
and
where we have used Lemma 2.
Hence, substituting (3.14) and (3.15) into (3.13) results in
which implies that
Clearly, we also have
Substituting (3.16) and (3.17) into (3.12) gives that
Multiplying both sides of (3.18) by \(2\tau \) and summing up the resulting equation, we have
An application of Gronwall inequality, we have
Substituting (3.20) into (3.16) yields that
Finally, by triangle inequality, one can check that
which is the desired result. The proof is complete.
4 Unconditionally superconvergent error estimate in \(H^1\)-norm of the fully-discrete scheme
We present the second main result in the following theorem.
Theorem 4.1
Suppose that \((u^n,\Phi ^n)\) and \((u_h^n,\Phi _h^n)\) are the solutions of (2.3)–(2.4) and (2.5)–(2.6) at \(t=t_n\), respectively. Moreover, suppose that \(u\in L^{\infty }(H^3(\Omega ))\), \(u_t, u_{tt}, u_{ttt}\in L^{\infty }(H^2(\Omega ))\), \(u_{tttt}\in L^{\infty }(L^2(\Omega ))\), \(\Phi \in L^{\infty }(H^3(\Omega ))\), \(\Phi _{tt}\in L^{\infty }(H^2(\Omega ))\), \(\Phi _{ttt}\in L^{\infty }(L^2(\Omega ))\). Then we have the following unconditionally superclose error estimate
where the constant C is independent of h, \(\tau \) and n, but depends on u, T.
Proof
Letting \(v_h=D_{\tau }\eta ^n\) in (3.2) and taking the real parts of the resulting equation give that
In terms of Cauchy–Schwarz inequality and (2.2), we have
Noticing that
we have from (3.1) that
and
where we have used (2.15), (3.1) and (3.21).
Hence, one can check that
In addition,by using Taylor expansion and integration by parts, we have
Substituting (4.3), (4.5) and (4.6) into (4.2) yields that
which implies that
In what follows, we pay our attention to estimate \(\Vert D_{\tau }\eta ^n\Vert _0\). To do this, subtracting the \(n-1\)-level from the n-level of (3.2), we have
Choosing \(v_h=D_{\tau }\bar{\eta }^n=\frac{1}{2}(D_{\tau }\eta ^n+D_{\tau }\eta ^{n-1})\) in (4.8) and taking the imaginary parts of the resulting equation, we have
By using Cauchy–Schwarz inequality, Taylor expansion and (2.2), we have
By using Cauchy–Schwarz inequality, Taylor expansion and integration by parts, we have
To estimate \(B_2\), we rewrite \((\bar{\Phi }^n\bar{u}^n-\bar{\Phi }_h^n\bar{u}_h^n)-(\bar{\Phi }^{n-1}\bar{u}^{n-1}-\bar{\Phi }_h^{n-1}\bar{u}_h^{n-1})\) as
According to Cauchy–Schwarz inequality, Taylor expansion and (3.1), it follows that
For \(B_2^2\), we have by (2.15) and (3.20)
For \(B_2^3\), there holds
In terms of (3.1), we have for \(B_2^4\) that
For \(B_2^5\), we have
By using (3.1), one can check that
To estimate the term \(-\tau (\bar{\theta }^{n-1}D_{\tau }\bar{\eta }^n,D_{\tau }\bar{\eta }^n)\) appeared on the right hand side of (4.17), we will discuss in two different cases.
Case I \(\tau \le h\). In this case, from (3.21), we have
which shows that
Hence, we conclude that
Case II \(\tau \ge h\). In this case, from (3.20), we have
Hence, we conclude that
where we have used (3.21).
Therefore, one can see that
Based on the estimates (4.18) and (4.23), we have
In addition, it follows that for \(B_2^6\)
Substituting the estimates \(B_2^1\sim B_2^6\) into \(B_2\), we have
Substituting the estimates \(B_1\sim B_6\) into (4.9) yields
Summing up the above inequality from 2 to n gives that
Next, we focus on the estimate \(\Vert \nabla D_{\tau }\theta ^n\Vert _0\). From (3.3), we have
Choosing \(w_h=D_{\tau }\theta ^n\) in (4.28) leads to
One can check that
By using Cauchy–schwarz inequality and (2.2), we have
It is not difficult to check that by (3.1)
For \(D_3\), we have by (2.15)
In addition, by (3.1), we have
By using (2.15) again, there holds
Thus, based on the above estimates \(D_1\sim D_5\), we conclude that
Substituting (4.36) into (4.27) results in
Finally, there remains the term \(\Vert D_{\tau }\eta ^1\Vert _0\) to estimate. To do this, letting \(n=1\) in (3.2),we have
where we have used \(\eta ^0=0\).
Choosing \(v_h=\frac{\eta ^1}{\tau }\) in (4.38) and taking the imaginary parts of the resulting equation give that
By using Cauchy–Schwarz inequality, (2.2), Taylor expansion and integration by parts, we have
Noticing that
we have by (2.15), (3.1) and (3.21)
Substituting (4.40) and (4.41) into (4.39) results in
Then, substituting (4.42) into (4.37) gives that
Hence, by (4.7) and (4.43), we have
An application of Gronwall inequality yields that
Furthermore, according to triangle inequality and the superclose estimate between \(R_hu^n\) and \(I_hu^n\) (Shi et al. 2014; Yang 2021), i.e., for \(u\in H^3(\Omega )\), there holds
Hence, we conclude that
Moreover, in terms of (3.21), we also have
The desired result (4.1) is obtained and the proof is complete.
In what follows, we adopt the interpolation post-processing approach to derive the global superconvergence result. A macroelement \(\widetilde{K}\) is constructed with 4 elements \(K_j\), \(j=1,2,3,4\) (see Fig. 1), the local interpolation operator \(I_{2\,h}: C(\widetilde{K})\rightarrow Q_{22}(\widetilde{K})\) is adopted as interpolation post-processing operator (Lin and Lin 2006) with the following interpolation conditions
where \(z_i\), \(i=1,2,\ldots ,9\) are the nine vertices of \(\widetilde{K}\) and \(Q_{22}(\widetilde{K})\) denotes biquadratic polynomial space on \(\widetilde{K}\).
What’s more, one can check that the properties, which have been shown in Lin and Lin (2006), for operator \(I_{2h}\) hold:
Therefore, in terms of (4.46) and (4.48)–(4.50), the global superconvergent error estiamte can be obtained.
Theorem 4.2
Under the conditions of Theorem 4.1, we have for \(n=1,2,\ldots ,N\)
Proof
From (4.48)–(4.50) and Theorem 4.1, one can see that
Similarly, we can derive the superconvergent result for \(\Phi ^n\). Hence, we complete the proof.
5 Numerical results
In this section, we present some numerical results to verify the correctness of the theoretical analysis.
Example 1
(Error estimates and order of convergence) We set the domain \(\Omega =(0,1)\times (0,1)\) and the final time \(T=1\) in the computation. Consider the following SP equation
Let the functions f and g and the initial and boundary conditions be chosen corresponding to the exact solutions
We present the numerical errors of \(\Vert u^n-u_h^n\Vert _0\), \(\Vert u^n-u_h^n\Vert _1\), \(\Vert I_hu^n-u_h^n\Vert _1\), \(\Vert u^n-I_{2\,h}u_h^n\Vert _1\) and \(\Vert \Phi ^n-\Phi _h^n\Vert _0\), \(\Vert \Phi ^n-\Phi _h^n\Vert _1\), \(\Vert I_h\Phi ^n-\Phi _h^n\Vert _1\), \(\Vert \Phi ^n-I_{2\,h}\Phi _h^n\Vert _1\) at \(t=0.2,~1.0\) in Tables 1, 2. Obviously, we can see that the numerical results agree well with the theoretical analysis, i.e., the convergence rates are \(O(h^2)\), O(h), \(O(h^2)\) and \(O(h^2)\), respectively.
Example 2
(Conservation of discrete mass and energy) We set the domain \(\Omega =(0,1)\times (0,1)\) and the final time \(T=100\). Consider the following SP equation
The temporal direction is divided with time stepsize \(\tau =1\), and the spatial direction is divided with stepsize \(h=\frac{\sqrt{2}}{40}\). In Fig. 2, we present some values of the discrete mass and energy for the scheme (2.5)–(2.6) at various time levels \(t^n\). It can be seen that the scheme (2.5)–(2.6) preserves the discrete mass and energy, which is consistent with the theoretical analysis.
Data Availability
Data will be made available on request.
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Acknowledgements
This work is supported by the National Natural Science Foundation of China (No. 12101568), the Key Scientific Research Projects Plan in Henan Higher Education Institutions (24A170031) and Henan Province General Project (242300421373), the Scientific Research Team Plan of Zhengzhou University of Aeronautics (23ZHTD01003).
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Yang, H., Liu, X. Unconditionally convergence and superconvergence error analysis of a mass- and energy-conserved finite element method for the Schrödinger–Poisson equation. Comp. Appl. Math. 43, 302 (2024). https://doi.org/10.1007/s40314-024-02822-3
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DOI: https://doi.org/10.1007/s40314-024-02822-3