An Error Analysis of the CN Weighed DG θ Method of the Convection Equation

: In this research paper, a weighed DG ﬁnite element method is proposed for solving convection equations with an easy execution and analysis. The key aim of this method is to design an error estimation for space and time of a discontinuous approximation on general ﬁnite element meshes. The efﬁciency of the parameter θ in the order of convergence of the solutions is also exposed. Some numerical examples were tested that demonstrated the strength and ﬂexibility of the method.


Introduction
Convection plays an important role in the application of meteorology, gas dynamics, oceanography, weather forecasting, aeroacoustics, turbulent flows, turbomachinery, oil recovery simulation, modeling of shallow water, the transport of contaminants in porous media, viscoelastic flows, magneto-hydrodynamics, and electro-magnetism. Recently, the discontinuous Galerkin (DG) method has been commonly used in industrial software packages for solving a large set of computational fluid problems. It is evident that it performs better than the standard continuous Galerkin method because of its flexibility in estimating, uneven solutions, mass conservation, unstructured meshes, potential for error control, and mesh adaptation. Among all FEMs, the DG method is a significant tool for solving convection equations. The calculation of the convection problem is simple, but it gives important numerical properties for more complicated problems. The discontinuous Galerkin finite element method (DGFEM) was introduced by Reed and Hill [1] in 1973 for solving the neutron transport equation. They compared their method with the continuous Galerkin finite element method by using numerical experiments and showed worthwhile stability properties of DGFEM. The first exploration of this method was published by Lesaint and Raviart [2] in 1974. Later on, this method was analyzed by using a stronger stability by Johnson, Navert, and J.Pitkäranta in [3] and Johnson and J.Pitkäranta in [4]. Jonson [5] gave an analysis of error control for stiff ODEs of the DG method in 1988, and then, Estep [6] in 1995 extended this analysis to general non-autonomous ODEs. Lastly, Böttcher and Rannacher [7] introduced a new adaptive error control method for ODEs by applying the DG method in 1996. On the other hand, Ricter in [8] explained the optimal rate of convergence for any structured two-dimensional non-Cartesian grid, and Lin, Yan, and Zhou in [9] exposed the first-order convergence of DG methods by using piece-wise constant approximations. After that, Lin in [10] reviewed this method and proposed a new error estimation for finite element approximations of hyperbolic equations. Süli [11] in 1996, Süli and Houston [12] in 1997, and Houston and Süli [13] in 2001 studied the a posteriori error analysis of hyperbolic problems. More recently, Houston, Schwab, and Süli in [14] widely studied the upwind finite element approximation for optimal control systems, while Xiong and Li in [15,16] established the a posteriori error estimation for the optimal control problems of the first-order linear hyperbolic and evolution equations, respectively. Then, Brezzi, Marini, and Süli in [17] adapted the upwind method by changing the standard upwind flux with a reliability term and a jump stabilization term. Again, Xiong and Li in [18] clarified the convergence properties of the optimal control problem governed by convection-diffusion equations. They extended the a posteriori error estimates and the a priori error estimates for both states, adjoints, and the control variable approximation. Furthermore, Xiong C., Luo F., and Ma X. et al. in [19] derived the a priori error analysis for the streamline diffusion discontinuous Galerkin finite element approximation of the optimal distributed control problem governed by the first-order linear hyperbolic equation. In this article, they analyzed the behavior of the error for h tending to zero and the degree p tending to infinity. Finally, Burman and Stamm in [20] verified that the optimal convergence holds for quadratic and higher polynomial degrees when the jump of the tangential part of the gradient is penalized. Mu and Ye in [21] introduced a simple method for solving firstorder hyperbolic equations of polygons/polyhedra of an arbitrary shape, and Wang in [22] presented the optimal L 2 error estimate of a linearized Crank-Nicolson Galerkin FEM for a generalized nonlinear Schrödinger equation without any time step size restriction. The least squares finite element solutions for linear hyperbolic problems were introduced in [23][24][25]. A significant literature exists on developing DG methods [26][27][28][29][30][31][32][33][34][35]. The objective of this article is to explore error estimates for both space and time of the convection equation by using the Crank-Nicolson discontinuous Galerkin (CN-DG) finite element method. The key aim of this method is to design the error estimation for both space and time on general finite element meshes. The reliability and effectiveness of the weighed parameter θ on the order of convergence of the solutions are also exposed. In the numerical experiments, we used the influence of the parameter θ in Figures 1,2, and 3 on the convergence of the solution. Most of the methods cited here established their approximation in a steady state without time discretization. Those methods (cited here) were considered time discretizations; their analysis was quite complicated and time consuming. It is also noteworthy to point out that it is a different and simpler approach to introduce such a parameter to seek an error analysis in time from all the DG schemes for convection equations. To the best of our knowledge, no proposals have been found in the literature that are relevant to such a type of parameter and penalty term in the error approximation of the convection equation. Therefore, we believe that the present way provides a simpler and more elegant error approximation of the convection equation for both space and time discretization.
The framework of the paper is as follows. The statement of the problem, other notations, and finite element meshes are introduced in Section 2. Moreover, we propose the main results, as well as the corresponding time-discrete scheme and give their detailed proofs in Section 3. Numerical examples are presented in Section 4 to demonstrate our theoretical results. Conclusions are drawn in Section 5.

Preliminaries
The purpose of this section is to state the problem in detail, introducing the finite element meshes and space and establishing some others preliminary results.

Formulation of the Problem
In this paper, a convection equation defined in a polyhedral bounded domain Ω in R d (d = 2, 3) is considered with the Lipschitz continuous boundary.
where the convection vector β = (β 1 , β 2 , β 3 , · · ·, β d ) T is a constant vector or a vector-valued function. Again, f is assumed to be a sufficiently smooth real-valued function, which could be different in different problems, and b is presumed as a constant. The initial u 0 is assumed to be bounded. The subsets of Γ = ∂Ω are defined as follows: where Γ − and Γ + are known as the inflow boundary and outflow boundary, respectively, and n(x) is the unit outward normal vector to Γ at x ∈ Γ. Here, Γ − and Γ + are not necessarily connected subsets of Ω. For simplicity, Ω is non-characteristic in the sense that Γ − Γ + . In addition, it is assumed that there exists a vector ξ ∈ R d such that, For the simplicity of (2), it is assumed ξ = 0 in the above hypothesis, and a positive constant b 0 is defined by:

Finite Element Mesh and Space
The approximation in the space of (1) is calculated by using discontinuous finite elements to construct finite element meshes over a polyhedron Ω. A family of partitions (T h ) h>0 is an affine shape-regular or one-irregular mesh sequence of the domain Ω. The mesh T h is composed of a combination of triangles T with the diameter h T . The mesh size is considered as h = max T∈T h h T . For each triangulation (T h ) h>0 , V h is introduced to denote the corresponding discontinuous finite element space of the piecewise polynomial P k (K) with the degree k ≥ 1.

CN-DG Approximation
In this paper, error estimations for both space and time were established for the convection Equation (1) by using the CN-DG finite element method. The purpose of this work is to find an approximation u n h ∈ V h such that: where u n = u(X, t n ), for a sequence of functions {u n } N n=0 , is defined, u n+ 1 2 = 1 2 (u n + u n+1 ), and δ + t u n = (u n+1 h − u n h ), and as usual, [u respectively, across the edges of an element over which it is calculated. {u . In this article, θ was used as a penalty parameter; the term (θβ · n[u (4) is called the penalty term. The penalty term is independent of time, convection terms, and source terms. The penalty parameter is only dependent on the mesh geometry of the sub-domain triangulation T h .

Main Results
The aspiration of this section is to introduce the stability of the CN-DG scheme, the error estimate, and the corresponding time-discrete error estimate. For the stability and error analysis, Gronwall inequalities are mostly required. For this reason, the following two lemmas are introduced. Lemma 1. (Gronwall inequality) Let y(t), g(t), h(t), and f (t) be nonnegative functions, and assume that there is a positive constant M independent of t such that t N 0 g(t)dt ≤ M and: Lemma 2. (Discrete Gronwall inequality) Let C and a n , b n , and d n , for integer 1 ≤ m ≤ t N k , be nonnegative numbers such that, Assume that kd n < 1 2 and ∀1 ≤ m ≤ t N k , then:

Stability of the CN-DG Scheme
To prove the stability of the scheme, the following DG-norm is defined by: Now, the following theorem is proposed for the stability analysis of the error estimate of the problem governed by Equation (1). Theorem 1. Assume that f ∈ C 2 (Ω × [0, T]) and u n h is the solution of (1), then there is a constant C > 0 independent of ∆t such that the following condition holds: Proof. Let V h = u n h + u n+1 h . By using Equation (4), it can be found that: Take the second and third parts of Equation (4), and apply the rules of integration: Again, consider the fourth and fifth parts of Equation (4): Now, combining the above Equations (6)-(8) and using Equation (4), it is found that, Now, by applying Equation (3), Insert Equations (10)-(12) into Equation (9).
By the Gronwall inequality, it is obtained:

Error Estimates
To describe the error estimate for the time-discrete system, the following theorem is proposed: Theorem 2. For any sequence {e n u } N n=0 and e n u = u(X, t n ) − u n , there is a constant C > 0 independent of ∆t such that the following condition holds: e n+1 u Proof. Let Equation (1) be represented as a time scheme in the following way: where, Now, denote e n u = u(X, t n ) − u n ; therefore, Multiplying by e n u + e n+1 u in the above equation, it is found that, By using the Gronwall inequality,

Spatial Error Estimate
To establish the main results, the following lemma is needed, which will play an important role in the error approximation. Lemma 3. Let us consider ∏ h to be an interpolation operator, then by the classical interpolation theory, For any two function u, v ∈ L 2 (Ω), define the inner product by: Let P : H 1 0 (Ω) → V h be a projection operator defined by: By the classical FEM theory, Now, the following theorem is proposed for the discussion about the main error analysis of the problem governed by Equation (1). Theorem 3. Let u n and u n h , n = 1, 2, ....., N denote the solutions of (1) and (4), respectively. For any 0 < τ ≤ T and h > 0, there is a constant C, which is dependent on τ h , but independent of u, τ 2 , and h r . Then, the following error representation formula holds: Proof. ξ n u = u n h − Pu n , and ρ n u = u n − Pu n . Therefore, using Equation (4), it is shown that: Let ; from Equation (16), it is found that: (βn[ξ Again, Combining Equations (17) to (21) and using the DG-norm defined in (5), we get, By using the Gronwall inequality above, ξ n+1 u 2 DG ≤ C(τ 2 + h r ). By using the triangle inequality, u n − u n h DG ≤ C(τ 2 + h r ).

Numerical Experiments
In this section, a number of numerical experiments are presented to confirm our theoretical analysis derived in Section 3. The mesh generation and all computations were done with FreeFem++ [36]. The error profiles and corresponding convergence rates are given in Tables 1-3. Consider the problem (4) on the circular domain Ω = {(x, y) ∈ R 2 : x 2 + y 2 ≤ 1} for all the numerical tests. The boundary condition and the proper function f and constant b were chosen for all the experiments such that u is the exact solution of (1). For all the experiments, the step size was considered as h = 1.0 mesh , and the time step was dt = h. The algorithm (4) was executed on the uniform triangular mesh sequence to authenticate the theoretical results. The discrete space V h was constructed by using piecewise polynomials of a uniform degree. The convergence history of the L 2 -norm of the error u n − u n h L 2 are displayed against the mesh-size. Figures 1-3 explain the behavior of the error u n h − u n L 2 at time t = 1 with respect to θ, while Figures 4-6 display the behavior of the error u n − u n h L 2 at time t = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} at a fixed h (mesh at 40) and θ . Furthermore, Figures 7-9 present the numerical results of u n h for the values 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 at time t = 0, 5, 7, 10.
Tables 1-3 present the L 2 -norm of the error u n − u n h L 2 and the order of convergence against the meshes at 5, 10, 20, 40, 80, and 160, respectively, at time t = 1. From all Tables 1-3, it is observed that the quantity u n − u n h L 2 has the convergence rates as predicted by the theorem, and the computed order of convergence is more than two.

Conclusions
In this article, the L 2 error estimation was developed by using the Crank-Nicolson discontinuous Galerkin finite element method. A spatial type of error estimate was designed for the fully discrete system, as well as the time-discrete system. Numerical experiments were presented that emphasized the reliability and effectiveness of the parameter θ on the optimal order of convergence of the solutions. The method used in this paper can also be extended to higher order schemes to obtain the L 2 error estimate.