The Crank–Nicolson finite element method for the 2D uniform transmission line equation

We develop the Crank–Nicolson finite element (CNFE) method for the two-dimensional (2D) uniform transmission line equation, study the stability and existence as well as error estimates for the CNFE solutions of the 2D uniform transmission line equation by strict theoretical approaches. We verify the correctness of the obtained theoretical results by means of numerical tests.


Introduction
Let Θ be an open bounded region in R 2 . We consider the uniform transmission line equation in the region Θ: where V represents the unknown voltage or current, V t = ∂V /∂t, V tt = ∂ 2 V /∂t 2 , V = ∂ 2 V /∂x 2 + ∂ 2 V /∂y 2 , F(x, y, t) represents the source term, V 0 (x, y, t) is the boundary value, H 0 (x, y) and H 1 (x, y) stand for initial values, T is the final moment, and σ = RG(ĈL) -1 > 0 and δ = (ĈR + LG)(ĈL) -1 > 0 due to G standing for the conductivity in the uniform transmission line, R for the impedance in the uniform transmission line, L for the inductance, andĈ for the electric capacity in the uniform transmission line. For convenience, we will premise that V 0 (x, y, t) = 0 in theoretical discussion. The uniform transmission line equation known as the telegraph equation is a crucial physical equation and has largely extensive applications (see [1,2]). Nevertheless, when its source term, boundary value, or initial values are complex, or its coefficients σ and δ are discontinuous, it has no genuine solution, so that we have to rely on numeric solutions.
The finite element (FE) method is one of most effective numerical methods and is used to solve many partial differential equations (see, e.g., [3][4][5][6]), whereas the Crank-Nicolson FE (CNFE) method has not been used to solve the uniform transmission line equation . Hence,  in this paper, we intend to develop the CNFE method for the 2D uniform transmission  line equation, analyze the stability and existence as well as errors for the CNFE solutions  of the 2D uniform transmission line equation, and verify the correctness of the obtained  theoretical results via numerical tests. Although the collocation spectral method, space-time finite element method, and finite difference scheme (see [7][8][9][10][11][12][13][14]) have been used to solve the uniform transmission line equation, they are different from the CNFE method, whereas the CNFE method has more merits than the methods mentioned; for example, the CNFE method for the 2D uniform transmission line equation is unconditionally stable, resulting in that it can ensure that the numerical solution is absolutely convergent and can obtain optimal order error estimates for the CNFE solutions unmatched for the above methods.
The paper is organized as follows. In Sect. 2, we firstly construct the CNFE model for the 2D uniform transmission line equation and analyze the existence, uniqueness, stability, and convergence of the CNFE solutions. Afterward, in Sect. 3, we use some numerical tests to verify the correctness of the obtained theoretical results. Lastly, we summarize the main conclusions and give some prospection in Sect. 4.

The CNFE method for the 2D uniform transmission line equation 2.1 The weak form of the 2D uniform transmission line equation
We will use the classical Sobolev spaces and norms (see, e.g., [15,16]). Let U = H 1 0 (Θ). Using Green's formula, we can derive the following weak form for the 2D uniform transmission line equation (1).
For Problem 1, we have the following conclusion of the existence, uniqueness, and stability of the weak solution.
Proof Since (2) is a linear equation system with respect to unknown function V , it has a unique solution if and only if it has merely a zero solution as F(x, y, t) = H 0 (x, y) = H 1 (x, y) = 0. Taking ϑ = V t in the first equation in Problem 1, we get Integrating (4) from 0 to t ∈ [0, T] and using the Hölder and Cauchy inequalities, we get Thereupon, when F(x, y, t) = H 0 (x, y) = H 1 (x, y) = 0, we obtain V 0 = ∇V 0 = 0, which implies u = 0, that is, the weak formulation (2) has a unique solution V ∈ H 1 0 (Θ). Further, from (5) we obtain (3). This completes the proof of Theorem 1.

The CNFE method for the 2D uniform transmission line equation
To solve Problem 1 by the CNFE method, let h be a uniformly regular triangulation on Θ (see, e.g., [16]). The FE subspace is defined as where . Then we can establish the following CNFE model for the 2D uniform transmission line equation.
where F(t n ) = F(x, y, t n ), and P h : U → U h is the Ritz projection (see [16]).
When g ∈ H l+1 (Θ)∩H 1 0 (Θ), the Ritz projection has the following boundedness and error estimates (see, e.g., [16]): where · 0 and · l represent, respectively, the norms in L 2 (Θ) and H l (Θ), and C > 0 stands for a generic constant independent of t and h, different at different places. Moreover, when g ∈ H 1 0 (Ω), it satisfies the Poincaré inequality For Problem 2, we have the existence, uniqueness, and stability of CNFE solutions.
, and H 1 ∈ H 1 (Θ), then Problem 2 has a unique sequence of solutions V n h ∈ U h (n = 1, 2, . . . , K ) satisfying the following unconditionally stability: Proof Since scheme (7) is a linear equation system with respect to V n+1 h , to prove that Problem 2 has a unique sequence of solutions, it suffices to check that it has merely zero solutions as in the first equation in (7) and applying the Hölder and Cauchy inequalities, we get Furthermore, we get Summing (11) from 1 to n, by the second formula in (7) and the Poincaré inequality we get for n = 1, 2, . . . , N -1. Thereupon, by (12) we have ∇V n h = V n h = 0 (n = 1, . . . , N ) as H 0 (x, y) = H 1 (x, y) = F(x, y, t) = 0, which shows that V n h = 0 (n = 1, 2, . . . , N ). Hence Problem 2 has a unique set of solutions. By (12) we directly get (9), which finishes the proof of Theorem 2.
The set of solutions for Problem 2 has the following error estimates.
Remark 1 The order of error estimates in Theorem 3 is optimal. Theorems 2 explains that the CNFE method (Problem 2) for the 2D uniform transmission line equation has a unique set of solutions that is unconditionally stable, so that it is unconditionally convergent and continuously depends on the initial value and source term. This shows that Problem 2 is theoretically reliable and effective for settling the 2D uniform transmission line equation.

Numerical tests
We employ some numerical tests to verify the correctness of the theoretical consequences of the CNFE method (Problem 2) of the 2D uniform transmission line equation, which can settle out the genuine solution, but usually it has no genuine solution.
In the 2D uniform transmission line equation (1), 1 and t ∈ [0, T)), H 0 (x, y) = 1 -cos 2πy cos 2πx, H 1 (x, y) = cos 2πy cos 2πx -1, and F(x, y, t) = 8π 2 exp(-t) cos 2πy cos 2πx. Thus we can find the following genuine solution for the uniform transmission line equation (1):  Table 1 shows the errors between the genuine solutions and the CNFE solutions and CPU run time for t = 0.3, t = 0.6, and t = 0.9, respectively, which verifies that the correctness of the theoretical results due to both theoretical and numerical errors reaching O(10 -4 ), signifies that the CNFE method is effective and reliable for calculating the 2D uniform transmission line equation. In Table 1, we also give the errors between the genuine and FE solutions and CPU run times when t = 0.3, t = 0.6, and t = 0.9 using the following FE method: where F(t n ) = F(x, y, t n ) and P h : U → U h is the Ritz projection. It is readily proven that Problem 3 is conditionally stable, so that it is conditionally convergent, and its FE solutions only have the first-order accuracy about time, that is, we have the following error estimates: Therefore, if t = 0.01 and l = 2 (also adopting second-degree elements), then the theoretical errors are of order O(10 2 ) only, which are consistent with numerical errors; see Table 1. It is further shown that the CNFE method (i.e., Problem 2) has more merits than the FE method (i.e., Problem 3) for settling the 2D uniform transmission line equation.

Conclusions and expectation
In our study, we have developed the CNFE method for the 2D uniform transmission line equation and analyzed the existence, uniqueness, stability, and errors for the CNFE solutions. We have also adopted some numeric tests to confirm the correctness for the CNFE method. It has been shown, by comparing with the usual FE method, that the CNFE method has more merits. Furthermore, it is revealed that the CNFE method is very effective for settling the 2D uniform transmission line equation.
In spite of the fact that we have only concerned with the CNFE method for the 2D uniform transmission line equation, the CNFE method may be applied to settle the threedimensional uniform transmission line equations or the uniform transmission line equation with complicated geometrical regions.
Although the CNFE method has many merits, such as unconditional stability, unconditional convergence, and the second-order accuracy in time, it has unknowns (freedom of degrees). When adopting the P 2 (K) with 2nd-degree polynomials on every triangular element in Sect. 3, we may reckon that the CNFE method has about 4 × 4 × 10 4 unknowns in the FE method (see [16,Lemma 1.30]). Provided that it is applied for settling big data processing in artificial intelligence and/or computational linguistics, there will be more than tens of millions unknown numbers. Fortunately, a proper orthogonal decomposition (POD) technique may be used to reduce the unknowns in the CNFE method, which can be used to reduce many numerical methods (see, e.g., [17][18][19][20][21][22]). Our future work is reducing the number of unknowns in the CNFE method by the POD technique.