Study of a New Mixed Weak Galerkin Formulation for the Electric Field

Abstract

In this paper, we present a new mixed weak Galerkin FEM for Maxwell’s equations in the primary electrostatic field–Lagrange multiplier. Our numerical scheme is equipped with stable finite elements composed of polynomials of degree ℓ for the electrostatic variable and polynomials of degree $𝓁+1$ for the Lagrange multiplier variable; the electrostatic field and the Lagrange multiplier variables are discontinuous. We demonstrate some error estimations that are optimal as a function of the mesh size and we study some numerical tests in a 2D domain. The numerical results perfectly confirm those shown theoretically.

1. Introduction

Recently, Wang and Ye [1] introduced a method for the resolution of partial differential equations, labeled the weak Galerkin finite element method. Several researchers have used the weak Galerin finite element method for approximating solutions of PDEs such as parabolic equations [2,3,4,5] and elliptic interface problems [6,7]. Recently, in [8], the authors introduce a new formulation based on WG-FEM, which discretizes the second order elliptic equation in non-mixed form directly. The mixed WG-FEM is an extension of the WG-FEM [9] and it was used for solving partial differential equations such as modified Maxwell’s equations [10] and Stokes problems [11,12,13,14]. There are recent numerical techniques for the resolution of Maxwell’s equations, for example, in [15], where the authors present a review on some recent progress achieved for simulating Maxwell’s equation, in [16], where an augmented mixed discontinuous Galerkin formulation for approximating the electric field is presented and analyzed, and in [17], the authors present a numerical study of conforming spacetime methods for Maxwell’s equations.

In this paper, we are interested to the following problem: Let $\Omega$ be a 2-dimensional bounded convex polygon or a 3-dimensional bounded polyhedral domain and $\Gamma$ its boundary, and then find the field u which satisfies

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \sigma u+\mathrm{curl}\left({\mu }^{-1}\mathrm{curl}u\right)-\epsilon \nabla p=& {\displaystyle J\mathrm{in}\Omega }\hfill \\ \hfill \mathrm{div}\left(\epsilon u\right)=& {\displaystyle 0\mathrm{in}\Omega }\hfill \\ \hfill n\wedge u=& {\displaystyle 0\mathrm{on}\Gamma .}\hfill \end{array}\end{array}$$

(1)

where $\Omega$ is a subset of ${\mathbb{R}}^d$, n is the unit outward on $\partial \Omega$, J is a current density, and $\mu$ and $\epsilon$ are the magnetic permeability and the electric permittivity of the medium, respectively. We assume that these coefficients are smooth, non-negative functions and in $L^{\infty }(\Omega )$. For a given non-zero frequency w, u is relied to the electric field E by the relation $E(x,t)=Re\left(u\right(x\left)\right(coswt+isinwt\left)\right)$. $\sigma =\sigma \left(x\right)\ge 0$ is any positive bounded function and may be considered zero almost in $\Omega$ and p is a Lagrange multiplier. For the case where $\sigma =0$, the term $\sigma u$ vanishes and the remaining problem is derived from Maxwell’s equations.

This paper is presented as follows: In following section, we introduce spaces and notations that are essential for the derivation of the WG formulation. Also, in this section, we present in detail our mixed weak Galerkin scheme. Section 3 discusses the properties of the bilinear forms defined in the numerical scheme and in Section 4, we analyze the convergence and prove some optimal error estimations. These theoretical convergence results were tested and confirmed numerically in Section 5.

2. Mixed FE Method

2.1. Notations and Meshes

Given a domain D, we keep notations for Sobolev spaces as in [18,19]. For the spaces $H(\mathrm{curl},D)$, $H({\mathrm{div}}_{\epsilon },D)$, and their subspaces, we refer to [20,21,22,23]. Assume that ${\mathcal{T}}_h$ is a subdivision of D into triangles if $D\subset {\mathbb{R}}^2$ and into tetrahedra if $D\subset {\mathbb{R}}^3$. We also assume that ${\mathcal{T}}_h$ is the regular shape and satisfies the hypothesis (A1)–(A4) given in [1]. Denote by ${\mathcal{E}}_I$ the set of all interior boundary elements of the partition and ${\mathcal{E}}_D$ the set of all exterior boundary elements of the partition and $\mathcal{E}={\mathcal{E}}_I\cup {\mathcal{E}}_D$. We consider the piecewise Sobolev spaces as

$$\begin{array}{c}\hfill H^s{\left({\mathcal{T}}_h\right)}^d:=\left\{u\in L^2{(\Omega )}^d:u_{|T}\in H^s{\left(T\right)}^d,\forall T\in {\mathcal{T}}_h\right\},d=1,2,3.\end{array}$$

2.2. Weak Galerkin Discretization

Let $𝓁\ge 0$, for an approximation of the electrostatic field

$$\begin{array}{c}\hfill {\mathcal{V}}_h=\left\{u=\{u^0,u^b\}:u^0\in {\left[P_{𝓁}\left(T\right)\right]}^d,u^b\in {\left[P_{𝓁}\left(e\right)\right]}^d,e\in \mathcal{E},T\in {\mathcal{T}}_h\right\}\end{array}$$

and its subspace

$$\begin{array}{ccc}\hfill {\mathcal{V}}_h^0& =& \left\{v\in {\mathcal{V}}_h:v^b\wedge n=0\mathrm{on}\Gamma \right\}.\hfill \end{array}$$

For the Lagrange multiplier, we approximate it in the finite element space

$$\begin{array}{c}\hfill {\mathcal{W}}_h=\left\{\psi \in L_0^2(\Omega ):{\psi }_{|K}\in P_{𝓁+1}\left(K\right),K\in {\mathcal{T}}_h\right\}.\end{array}$$

Now, let us define the weak operators. For $K\in {\mathcal{T}}_h$ and $v\in {\mathcal{V}}_h$ as two arbitrary elements, we denote ${\mathrm{curl}}_{w}v\in {\left[P_{𝓁}\left(K\right)\right]}^d$ the weak curl differential operator as the only polynomial, satisfying

$$\begin{array}{c}\hfill {({\mathrm{curl}}_{w}v,w)}_K={(v^0,\mathrm{curl}w)}_K-{⟨v^b\wedge n,w⟩}_{\partial K},\forall w\in {\left[P_{𝓁}\left(K\right)\right]}^d.\end{array}$$

(2)

A weak divergence ${\mathrm{div}}_{w}v\in P_{𝓁+1}\left(K\right)$ is considered as the only polynomial such that

$$\begin{array}{c}\hfill {({\mathrm{div}}_{w}v,\psi )}_K=-{(v^0,\nabla \psi )}_K+{⟨v^b·n,\psi ⟩}_{\partial K},\forall \psi \in P_{𝓁+1}\left(K\right).\end{array}$$

(3)

With the previous notations and the last definitions of weak differential operators, we can consider an approximation of (1) as follows: Find $(u_h,p_h)\in {\mathcal{V}}_h^0\times {\mathcal{W}}_h$, satisfying

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill {\displaystyle \sum _{K\in {\mathcal{T}}_h}{(\sigma u_h,v_h)}_K+{({\mu }^{-1}{\mathrm{curl}}_w{u}_h,{\mathrm{curl}}_w{v}_h)}_K}& +\hfill & {\displaystyle {({\mathrm{div}}_w\epsilon u_h,{\mathrm{div}}_w\epsilon v_h)}_K+{(p_h,{\mathrm{div}}_w\epsilon v_h)}_K}\hfill \\ & =\hfill & {\displaystyle \sum _{K\in {\mathcal{T}}_h}{(J,v_h^0)}_K,\forall v_h\in {\mathcal{V}}_h^0,}\hfill \\ \hfill {\displaystyle \sum _{K\in {\mathcal{T}}_h}{({\mathrm{div}}_w\epsilon u_h,{\psi }_h)}_K}& =\hfill & {\displaystyle 0,\forall {\psi }_h\in {\mathcal{W}}_h.}\hfill \end{array}\end{array}$$

(4)

Clearly, this last system is no longer consistent due to an insufficient enforcement of the components $u_h^0$ and $u_h^b$. Therefore, some stabilization terms must be added and we stabilize the bilinear form

$$\sum _{K\in {\mathcal{T}}_h}{(\sigma u_h,v_h)}_K+{({\mu }^{-1}{\mathrm{curl}}_w{u}_h,{\mathrm{curl}}_w{v}_h)}_K+{({\mathrm{div}}_w\epsilon u_h,{\mathrm{div}}_w\epsilon v_h)}_K$$

by requiring some communications between $u_h^0$ and $u_h^b$. Hence, we introduce the following bilinear forms:

$$\begin{array}{ccc}\hfill a(u_h,v_h)& :=& \sum _{K\in {\mathcal{T}}_h}{(\sigma u_h,v_h)}_K+{({\mu }^{-1}{\mathrm{curl}}_w{u}_h,{\mathrm{curl}}_w{v}_h)}_K+{({\mathrm{div}}_w\epsilon u_h,{\mathrm{div}}_w\epsilon v_h)}_K,\hfill \\ \hfill B(v_h,p_h)& :=& \sum _{K\in {\mathcal{T}}_h}{(p_h,{\mathrm{div}}_w\epsilon v_h)}_K(v_h,p_h)\in {\mathcal{V}}_h^0\times {\mathcal{W}}_h,\hfill \\ \hfill s_K(u_h,v_h)& :=& {\displaystyle \delta \sum _{\partial K\in {\mathcal{E}}_h^I}{h}_K^{-1}{⟨(\epsilon u_h^0-\epsilon u_h^b)·n,(\epsilon v_h^0-\epsilon v_h^b)·n⟩}_{\partial K}}\hfill \\ & & +\delta \sum _{\partial K\in {\mathcal{E}}_h}{h}_K^{-1}{⟨(u_h^0-u_h^b)\wedge n,(v_h^0-v_h^b)\wedge n⟩}_{\partial K}(u_h,v_h)\in {\mathcal{V}}_h^0\times {\mathcal{V}}_h^0,\hfill \end{array}$$

where $\delta >0$ is an arbitrary real parameter. Now, we consider an approximation of (1) as follows: Find $u_h=\{u_h^0,u_h^b\}\in {\mathcal{V}}_h^0$ and $p_h\in {\mathcal{W}}_h$, satisfying

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill A_s(u_h,v_h)+B(v_h,p_h)=& {\displaystyle \sum _{K\in {\mathcal{T}}_h}(J,v_h^0),\forall v_h\in {\mathcal{V}}_h^0,}\hfill \\ \hfill B(u_h,{\psi }_h)=& {\displaystyle 0,\forall {\psi }_h\in {\mathcal{W}}_h}\hfill \end{array}\end{array}$$

(5)

with

$$\begin{array}{c}\hfill A_s(u_h,v_h)=a(u_h,v_h)+s_K(u_h,v_h)(u_h,v_h)\in {\mathcal{V}}_h^0\times {\mathcal{V}}_h^0.\end{array}$$

3. Solvability and Stability

Let us start by introducing the local projection operators. Define ${\mathbb{Q}}_0:{\left(L^2\left(T\right)\right)}^d\to {\left(P_k\left(T\right)\right)}^d$ and ${\mathbb{Q}}_b:{\left(L^2\left(e\right)\right)}^d\to {\left(P_k\left(e\right)\right)}^d$ as the local projection operators on every elements T of ${\mathcal{T}}_h$ and e of ${\mathcal{E}}_h$, respectively. ${\mathbb{Q}}_h$ is the $L^2$-projection of $v=\{v^0,v^b\}\in {\mathcal{V}}_h^0$ defined as ${\mathbb{Q}}_h=\{{\mathbb{Q}}_0\left(v^0\right),{\mathbb{Q}}_b\left(v^b\right)\}.$ For $p\in {\mathcal{W}}_h$, we denote by $Q_h\left(p\right):L^2\left(T\right)\to P_{k+1}\left(T\right)$ its local projection operator from $L^2\left(T\right)$. In the following part of this section, we study the properties of bilinear forms. We start by defining norms, which are necessary for our study. For $\psi \in {\mathcal{W}}_h$, we use the usual $L^2-$ norm $\parallel \psi \parallel$, while for $q\in {\mathcal{V}}_h^0$, we define the norm of q as

$$\begin{array}{c}\hfill {\left|\left|\left|q\right|\right|\right|}^2:=A_s(q,q).\end{array}$$

Using the classical techniques and the last definition of $\left|\left|\left|·\right|\right|\right|$, we can show that $A_s$ and B are continuous on spaces ${\mathcal{V}}_h^0\times {\mathcal{V}}_h^0$ and ${\mathcal{V}}_h^0\times {\mathcal{W}}_h$, respectively, while the coercivity of $A_s(v,v)$ on ${\mathcal{V}}_h^0\times {\mathcal{V}}_h^0$ follows from the definition of the triple bar norm $A_s(w,w)={\left|\left|\left|w\right|\right|\right|}^2$. Therefore, for an application of the Babuška–Brezzi theory, an inf-sup condition for B remains to be shown.

Lemma 1.

We have

$$\begin{array}{c}\hfill \underset{\psi \in {\mathcal{W}}_h\setminus \left\{0\right\}}{inf}\underset{v\in {\mathcal{V}}_h^0\setminus \left\{0\right\}}{sup}{\displaystyle \frac{B(\phi ,v)}{\left|\left|\left|v\right|\right|\right|\parallel \phi \parallel }}\ge \varrho >0\end{array}$$

(6)

with a positive constant ϱ independent of h.

Proof. 

Fix $\psi$ as an element of ${\mathcal{W}}_h$, then $\psi \in L_0^2(\Omega )$ and we know [13,24] that there exists v in $H_0^1{(\Omega )}^d$ with $(\mathrm{div}v,\psi )\ge {C\parallel \psi \parallel \parallel v\parallel }_1.$ Let $\tilde{v}={\mathbb{Q}}_h\left(v\right)$ and let us prove that $\left|\left|\left|\tilde{v}\right|\right|\right|\le C{\parallel v\parallel }_1$. Using the fact that $\sigma$ is bounded, one can obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \sum _{T\in {\mathcal{T}}_h}{\parallel \sigma \tilde{v}\parallel }^2=}& \sum _{T\in {\mathcal{T}}_h}{\parallel \sigma {\mathbb{Q}}_h\left(v\right)\parallel }^2\hfill \\ \hfill \le & {\displaystyle C\sum _{T\in {\mathcal{T}}_h}{\parallel {\mathbb{Q}}_h\left(v\right)\parallel }^2}\hfill \\ \hfill \le & {\displaystyle C\sum _{T\in {\mathcal{T}}_h}{\parallel v\parallel }^2\le {\parallel v\parallel }_1^2.}\hfill \end{array}\end{array}$$

(7)

For an estimation of the norms of the weak differential operators ${\sum }_{K\in {\mathcal{T}}_h}{\parallel {\mathrm{curl}}_w\tilde{v}\parallel }^2$ and ${\sum }_{K\in {\mathcal{T}}_h}{\parallel {\mathrm{div}}_w\tilde{v}\parallel }^2$, as in [13], we have shown that

$$\begin{array}{ccc}\hfill {\mathrm{div}}_w{\left({\mathbb{Q}}_h\left(u\right)\right)}_K& =& Q_h{\left(\mathrm{div}u\right)}_K,\hfill \\ \hfill {\mathrm{curl}}_w{\left({\mathbb{Q}}_h\left(u\right)\right)}_K& =& {\mathbf{Q}}_h{\left(\mathrm{curl}u\right)}_K.\hfill \end{array}$$

Then, we write

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill {\displaystyle \sum _{T\in {\mathcal{T}}_h}\parallel {\mathrm{curl}}_w\tilde{v}{\parallel }^2=\sum _{T\in {\mathcal{T}}_h}\parallel {\mathrm{curl}}_w{\mathbb{Q}}_h{\left(v\right)\parallel }^2=\sum _{T\in {\mathcal{T}}_h}{\parallel {\mathbf{Q}}_h\left(\mathrm{curl}v\right)\parallel }^2}& \le \hfill & {\displaystyle \sum _{T\in {\mathcal{T}}_h}{\parallel \mathrm{curl}v\parallel }^2}\hfill \\ & \le \hfill & {\displaystyle {\parallel v\parallel }_1^2}\hfill \end{array}\end{array}$$

(8)

and

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill {\displaystyle \sum _{T\in {\mathcal{T}}_h}\parallel {\mathrm{div}}_w\tilde{v}{\parallel }^2=\sum _{T\in {\mathcal{T}}_h}\parallel {\mathrm{div}}_w{\mathbb{Q}}_h{\left(v\right)\parallel }^2=\sum _{T\in {\mathcal{T}}_h}{\parallel Q_h\left(\mathrm{div}v\right)\parallel }^2}& \le \hfill & {\displaystyle \sum _{T\in {\mathcal{T}}_h}{\parallel \mathrm{div}v\parallel }^2}\hfill \\ & \le \hfill & {\displaystyle {\parallel v\parallel }_1^2.}\hfill \end{array}\end{array}$$

(9)

Now, for an estimation of $s_K(\tilde{v},\tilde{v})$, introduce the notations ${\mathbb{Q}}_b^v\left(v\right)={\mathbb{Q}}_b\left(v\right)-v$ and ${\mathbb{Q}}_0^v\left(v\right)={\mathbb{Q}}_0\left(v\right)-v$ and remark that

$$\begin{array}{ccc}\hfill {\parallel {\mathbb{Q}}_b^v\left(v\right)\parallel }_{\partial K}^2& =& {⟨{\mathbb{Q}}_b^v\left(v\right),{\mathbb{Q}}_b^v\left(v\right)⟩}_{\partial K}\hfill \\ & =& {⟨{\mathbb{Q}}_b^v\left(v\right),{\mathbb{Q}}_0^v\left(v\right)⟩}_{\partial K}\hfill \\ & \le & {\parallel {\mathbb{Q}}_b^v\left(v\right)\parallel }_{\partial K}{\parallel {\mathbb{Q}}_0^v\left(v\right)\parallel }_{\partial K}\hfill \end{array}$$

and then

$$\begin{array}{c}\hfill {\parallel {\mathbb{Q}}_b^v\left(v\right)\parallel }_{\partial K}\le {\parallel {\mathbb{Q}}_0^v\left(v\right)\parallel }_{\partial K}.\end{array}$$

(10)

Therefore

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill {\displaystyle }& \hfill & {\displaystyle \delta \sum _{\partial K\in {\mathcal{E}}_h}{h}_K^{-1}{\parallel ({\tilde{v}}^0-{\tilde{v}}^b)\wedge n\parallel }_{\partial K}^2}\hfill \\ & =\hfill & {\displaystyle \delta \sum _{\partial K\in {\mathcal{E}}_h}{h}_K^{-1}{\parallel ({\mathbb{Q}}_0\left(v\right)-{\mathbb{Q}}_b\left(v\right))\wedge n\parallel }_{\partial K}^2}\hfill \\ & \le \hfill & {\displaystyle 2\delta \sum _{\partial K\in {\mathcal{E}}_h}{h}_K^{-1}\parallel {\mathbb{Q}}_0^v{\left(v\right))\parallel }_{\partial K}^2+2\delta \sum _{\partial K\in {\mathcal{E}}_h}{h}_K^{-1}{\parallel {\mathbb{Q}}_b^v\left(v\right)\parallel }_{\partial K}^2}\hfill \\ & \le \hfill & {\displaystyle 2\delta \sum _{\partial K\in {\mathcal{E}}_h}{h}_K^{-1}\parallel {\mathbb{Q}}_0^v{\left(v\right)\parallel }_{\partial K}^2+2\delta \sum _{\partial K\in {\mathcal{E}}_h}{h}_K^{-1}{\parallel {\mathbb{Q}}_0^v\left(v\right)\parallel }_{\partial K}^2}\hfill \\ & \le \hfill & {\displaystyle C\delta \sum _{\partial K\in {\mathcal{E}}_h}{h}_K^{-1}{\parallel {\mathbb{Q}}_0^v\left(v\right)\parallel }_{\partial K}^2}\hfill \\ & \le \hfill & {\displaystyle C\delta \sum _{K\in {\mathcal{T}}_h}{h}_K^{-2}\parallel {\mathbb{Q}}_0^v{\left(v\right)\parallel }_T^2+{\parallel \nabla \left({\mathbb{Q}}_0^v\left(v\right)\right)\parallel }_T^2}\hfill \\ & \le \hfill & {\displaystyle C\delta \sum _{K\in {\mathcal{T}}_h}{h}_K^{-2}\parallel {\mathbb{Q}}_0^v{\left(v\right)\parallel }_T^2+{\parallel \nabla \left({\mathbb{Q}}_0^v\left(v\right)\right)\parallel }_T^2}\hfill \\ & \le \hfill & {\displaystyle {C\parallel \nabla v\parallel }^2.}\hfill \end{array}\end{array}$$

(11)

Similarly, one can obtain

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\displaystyle \delta \sum _{\partial K\in {\mathcal{E}}_h}{h}_K^{-1}\parallel ({\tilde{v}}^0-{\tilde{v}}^b){·n\parallel }_{\partial K}^2\le C{\parallel \nabla v\parallel }^2.}\end{array}\end{array}$$

(12)

From (7), (8), (9), (11), and (12), we deduce that

$$\begin{array}{c}\hfill \left|\left|\left|\tilde{v}\right|\right|\right|\le C{\parallel v\parallel }_1.\end{array}$$

From the definition of $Q_h$ and Lemma 2 given in next section, we obtain

$$\begin{array}{c}\hfill B(\tilde{v},\psi )=({\mathrm{div}}_w{\mathbb{Q}}_h\left(v\right),\psi )=(Q_h\left(\mathrm{div}v\right),\psi )=(\mathrm{div}v,\psi )\ge {C\parallel v\parallel }_1\parallel \psi \parallel \ge \varrho \left|\left|\left|\tilde{v}\right|\right|\right|\parallel \psi \parallel .\end{array}$$

□

4. Error Estimate

Let us first recall the trace inequality [9], for every K in ${\mathcal{T}}_h$ and for any e in ${\mathcal{E}}_h$, one has

$$\begin{array}{c}\hfill {\parallel \theta \parallel }_e^2\le C\left(h_K^{-1}{\parallel \phi \parallel }_K^2+h_K{\parallel \nabla \theta \parallel }_K^2\right)\forall \theta \in H^1\left(K\right).\end{array}$$

(13)

Now, we introduce the next Lemma, which plays a paramount role in our study of error estimations.

Lemma 2.

Assume that u and p satisfy (1), then

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill {\mathrm{div}}_w{\left({\mathbb{Q}}_h\left(u\right)\right)}_K& =\hfill & Q_h{\left(\mathrm{div}u\right)}_K,\hfill \\ \hfill {\mathrm{curl}}_w{\left({\mathbb{Q}}_h\left(u\right)\right)}_K& =\hfill & {\mathbf{Q}}_h{\left(\mathrm{curl}u\right)}_K,\hfill \\ \hfill {({\mathrm{curl}}_{w}v,{\mathbb{Q}}_h\left(\phi \right))}_K& =\hfill & {(v^0,\mathrm{curl}\phi )}_K+{⟨(v^b-v^0)\wedge n,\phi -{\mathbb{Q}}_h\left(\phi \right)⟩}_{\partial K}-{⟨v^b\wedge n,\phi ⟩}_{\partial K}.\hfill \end{array}\end{array}$$

Proof. 

The proof follows similar techniques given in [13]. □

4.1. Error Equations

Our objective here is to demonstrate some error equations which are necessary for establishing optimal error estimation for our mixed weak Galerkin discrete scheme (5). In order to simplify the calculations, we assume that parameters $\mu$, $\epsilon$ are constants and to be congruent to the identity, so $\epsilon \left(x\right)\equiv \mu \left(x\right)\equiv 1$ almost in $\Omega$. $(u,p)$ is a solution of (1), which is assumed to be sufficiently regular. Using Lemma 2, Equation (2), and integrating parts, we can arrive to

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill {\displaystyle {({\mathrm{curl}}_w\left({\mathbb{Q}}_h\left(u\right)\right),{\mathrm{curl}}_{w}v)}_K}& =\hfill & {\displaystyle {({\mathbf{Q}}_h\left(\mathrm{curl}u\right),{\mathrm{curl}}_{w}v)}_K}\hfill \\ & =\hfill & {\displaystyle {(v^0,\mathrm{curl}\left({\mathbf{Q}}_h\left(\mathrm{curl}u\right)\right))}_K-{⟨v^b\wedge n,{\mathbf{Q}}_h\left(\mathrm{curl}u\right)⟩}_{\partial K}}\hfill \\ & =\hfill & {\displaystyle {(\mathrm{curl}{v}^0,{\mathbf{Q}}_h\left(\mathrm{curl}u\right))}_K-{⟨(v^b-v^0)\wedge n,{\mathbf{Q}}_h\left(\mathrm{curl}u\right)⟩}_{\partial K}}\hfill \\ & =\hfill & {\displaystyle {(\mathrm{curl}{v}^0,\mathrm{curl}u)}_K-{⟨(v^b-v^0)\wedge n,{\mathbf{Q}}_h\left(\mathrm{curl}u\right)⟩}_{\partial K}.}\hfill \end{array}\end{array}$$

(14)

Also from Equation (3), the classical integration of parts based on the formula and since ${\sum }_{K\in {\mathcal{T}}_h}{⟨v^b·n,p⟩}_{\partial K}$ is null, we obtain (see [13])

$$\begin{array}{c}\hfill -{({\mathrm{div}}_{w}v,Q_h\left(p\right))}_{\Omega }={(v^0,\nabla p)}_{\Omega }+\sum _{K\in {\mathcal{T}}_h}{⟨(v^b-v^0)·n,p-Q_h\left(p\right)⟩}_{\partial K}.\end{array}$$

Therefore

$$\begin{array}{c}\hfill (v^0,\nabla p)=-{({\mathrm{div}}_{w}v,Q_h\left(p\right))}_{\Omega }-\sum _{K\in {\mathcal{T}}_h}{⟨(v^b-v^0)·n,p-Q_h\left(p\right)⟩}_{\partial K}.\end{array}$$

(15)

Now, multiplying the first equation of (1) with $v^0$ in $v=\{v^0,v^b\}\in V_h^0$, we obtain

$$\begin{array}{c}\hfill {(\sigma u,v^0)}_{\Omega }+{(\mathrm{curl}\left(\mathrm{curl}u\right),v^0)}_{\Omega }-{(\nabla p,v^0)}_{\Omega }={(J,v^0)}_{\Omega }.\end{array}$$

(16)

The addition of the term ${(\sigma {\mathbb{Q}}_h\left(u\right),v^0)}_{\Omega }$ to (16) means

$$\begin{array}{c}\hfill {(\sigma {\mathbb{Q}}_h\left(u\right),v^0)}_{\Omega }+{(\mathrm{curl}\left(\mathrm{curl}u\right),v^0)}_{\Omega }-{(\nabla p,v^0)}_{\Omega }={(\sigma ({\mathbb{Q}}_h\left(u\right)-u),v^0)}_{\Omega }+{(J,v^0)}_{\Omega }.\end{array}$$

(17)

Since ${\sum }_{K\in {\mathcal{T}}_h}{⟨v^b\wedge n,\left(\mathrm{curl}u\right)⟩}_{\partial K}=0$ and using an integration by parts, it is straightforward to obtain after an integration by parts and (14)

$$\begin{array}{ccc}\hfill {(\mathrm{curl}\left(\mathrm{curl}u\right),v^0)}_{\Omega }& =& \sum _{K\in {\mathcal{T}}_h}{(\mathrm{curl}u,\mathrm{curl}{v}^0)}_K\hfill \\ & & -\sum _{K\in {\mathcal{T}}_h}{⟨(v^b-v^0)\wedge n,\left(\mathrm{curl}u\right)⟩}_{\partial K}\hfill \\ & =& {({\mathrm{curl}}_w\left({\mathbb{Q}}_h\left(u\right)\right),{\mathrm{curl}}_{w}v)}_{\Omega }\hfill \\ & & -\sum _{K\in {\mathcal{T}}_h}{⟨n\wedge (v^b-v^0),{\mathbf{Q}}_h\left(\mathrm{curl}u\right)⟩}_{\partial K}\hfill \\ & & +\sum _{K\in {\mathcal{T}}_h}{⟨n\wedge (v^b-v^0),\left(\mathrm{curl}u\right)⟩}_{\partial K}\hfill \\ & =& {(\sigma ({\mathbb{Q}}_h\left(u\right)-u),v^0)}_{\Omega }+{({\mathrm{curl}}_w\left({\mathbb{Q}}_h\left(u\right)\right),{\mathrm{curl}}_{w}v)}_{\Omega }\hfill \\ & & +\sum _{K\in {\mathcal{T}}_h}{⟨(v^b-v^0)\wedge n,{\mathbf{Q}}_h\left(\mathrm{curl}u\right)-\mathrm{curl}u⟩}_{\partial K}.\hfill \end{array}$$

The last equation and (15) substituted into (17) gives

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill (\sigma {\mathbb{Q}}_h\left(u\right),v^0)& +\hfill & {({\mathrm{curl}}_w\left({\mathbb{Q}}_h\left(u\right)\right),{\mathrm{curl}}_{w}v)}_{\Omega }+({\mathrm{div}}_{w}v,Q_h\left(p\right))\hfill \\ & =\hfill & {(\sigma ({\mathbb{Q}}_h\left(u\right)-u),v^0)}_{\Omega }+(J,v^0)\hfill \\ & \hfill & {\displaystyle -\sum _{K\in {\mathcal{T}}_h}{⟨(v^b-v^0)\wedge n,{\mathbf{Q}}_h\left(\mathrm{curl}u\right)-\mathrm{curl}u⟩}_{\partial K}}\hfill \\ & \hfill & {\displaystyle -\sum _{K\in {\mathcal{T}}_h}{⟨(v^b-v^0)·n,p-Q_h\left(p\right)⟩}_{\partial K}.}\hfill \end{array}\end{array}$$

(18)

Now, multiply the second equation in (1) with ${\mathrm{div}}_{w}v$ and obtain

$$\begin{array}{c}\hfill 0=(\mathrm{div}u,\mathrm{div}v)=(Q_h\left(\mathrm{div}u\right),{\mathrm{div}}_{w}v)=({\mathrm{div}}_w\left({\mathbb{Q}}_h\left(u\right)\right),{\mathrm{div}}_{w}v).\end{array}$$

which means

$$\begin{array}{c}\hfill {({\mathrm{div}}_w\left({\mathbb{Q}}_h\left(u\right)\right),{\mathrm{div}}_{w}v)}_{\Omega }=0\end{array}$$

(19)

and it follows after the addition of Equations (18) and (19) that

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill (\sigma {\mathbb{Q}}_h\left(u\right),v^0)& +\hfill & ({\mathrm{curl}}_w\left({\mathbb{Q}}_h\left(u\right)\right),{\mathrm{curl}}_{w}v)+({\mathrm{div}}_w\left({\mathbb{Q}}_h\left(u\right)\right),{\mathrm{div}}_{w}v)+({\mathrm{div}}_{w}v,Q_h\left(p\right))\hfill \\ & =\hfill & {\displaystyle (\sigma ({\mathbb{Q}}_h\left(u\right)-u),v^0)+(J,v^0)}\hfill \\ & \hfill & {\displaystyle -\sum _{K\in {\mathcal{T}}_h}{⟨(v^b-v^0)\wedge n,{\mathbf{Q}}_h\left(\mathrm{curl}u\right)-\mathrm{curl}u⟩}_{\partial K}}\hfill \\ & \hfill & {\displaystyle -\sum _{K\in {\mathcal{T}}_h}{⟨(v^b-v^0)·n,p-Q_h\left(p\right)⟩}_{\partial K}.}\hfill \end{array}\end{array}$$

(20)

Now, it is time to introduce and prove the error equations.

Lemma 3.

We have

$$\begin{array}{ccc}\hfill A_s(e_h,v)+B(v,{ϵ}_h)& =& {(\sigma ({\mathbb{Q}}_h\left(u\right)-u),v^0)}_{\Omega }+\sum _{K\in {\mathcal{T}}_h}{⟨n\wedge (v^0-v^b),{\mathbf{Q}}_h\left(\mathrm{curl}u\right)-\mathrm{curl}u⟩}_{\partial K}\hfill \\ & & -s_K({\mathbb{Q}}_h\left(u\right),v)-\sum _{K\in {\mathcal{T}}_h}{⟨(v^0-v^b)·n,p-Q_h\left(p\right)⟩}_{\partial K},\hfill \\ \hfill B(e_h,\psi )& =& 0\hfill \end{array}$$

with ${ϵ}_h:=p_h-Q_h\left(p\right)$ and $e_h:=u_h-{\mathbb{Q}}_h\left(u\right)$ are the errors.

Proof. 

The addition of $s_K({\mathbb{Q}}_h\left(u\right),v)$ to (20) implies

$$\begin{array}{ccc}\hfill A_s({\mathbb{Q}}_h\left(u\right),v)+B(v,Q_h\left(p\right))& =& (\sigma ({\mathbb{Q}}_h\left(u\right)-u),v^0)+(J,v^0)\hfill \\ & & +\sum _{K\in {\mathcal{T}}_h}{⟨n\wedge (v^b-v^0),{\mathbf{Q}}_h\left(\mathrm{curl}u\right)-\mathrm{curl}u⟩}_{\partial K}\hfill \\ & & +s_K({\mathbb{Q}}_h\left(u\right),v)-\sum _{K\in {\mathcal{T}}_h}{⟨(v^b-v^0)·n,p-Q_h\left(p\right)⟩}_{\partial K}.\hfill \end{array}$$

We subtract this last equation from the first equation in (5) and obtain

$$\begin{array}{ccc}\hfill A_s(e_h,v)+B(v,{ϵ}_h)& =& {(\sigma ({\mathbb{Q}}_h\left(u\right)-u),v^0)}_{\Omega }+\sum _{K\in {\mathcal{T}}_h}{⟨n\wedge (v^0-v^b),{\mathbf{Q}}_h\left(\mathrm{curl}u\right)-\mathrm{curl}u⟩}_{\partial K}\hfill \\ & & -s_K({\mathbb{Q}}_h\left(u\right),v)-\sum _{K\in {\mathcal{T}}_h}{⟨(v^0-v^b)·n,p-Q_h\left(p\right)⟩}_{\partial K}.\hfill \end{array}$$

Now, we multiply the second equation in (1) by $\psi$, then

$$\begin{array}{c}\hfill 0=(\mathrm{div}u,\psi )=(Q_h\left(\mathrm{div}u\right),\psi )=({\mathrm{div}}_w\left({\mathbb{Q}}_h\left(u\right)\right),\psi ),\end{array}$$

(21)

which means

$$\begin{array}{c}\hfill B({\mathbb{Q}}_h\left(u\right),\psi )=0.\end{array}$$

This last equation subtracted from the second equation in (5) implies

$$\begin{array}{c}\hfill B(e_h,\psi )=0.\end{array}$$

□

4.2. Error Estimations

In the following theorem, we prove some optimal order error estimations for the electrostatic variable $u_h$ and the Lagrange multiplier variable $p_h$ in the $\left|\left|\left|·\right|\right|\right|$ norm and the standard $L^2$ norm, respectively. Also, we give an $L^2$ error estimation result for the electrostatic field $u_h$.

Theorem 1.

Assume that u and p satisfy (1) such that u in $H^{k+2}(\Omega )$ and p in $H^{k+1}(\Omega )$ for $k\ge 0$. Let $(u_h,p_h)\in {\mathcal{V}}_h^0\times {\mathcal{W}}_h$, satisfying (5). Then

$$\begin{array}{c}\hfill \left|\left|\left|{\mathbb{Q}}_h\left(u\right)-u_h\right|\right|\right|+\parallel Q_h\left(p\right)-p_h\parallel \le C{h}^{s+1}\left[{\parallel u\parallel }_{s+2}+{\parallel p\parallel }_{s+1}\right]\end{array}$$

and

$$\begin{array}{c}\hfill \parallel {\mathbb{Q}}_0\left(u\right)-u_0\parallel \le C{h}^{k+2}\left[{\parallel u\parallel }_{k+2}+{\parallel p\parallel }_{k+1}\right].\end{array}$$

Proof. 

Let us begin by introducing the two inequality types valid for $u\in H^{𝓁+2}(\Omega )$ and $s\in [0,𝓁+1]$; we refer to [9] for the detail of their proofs.

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill {\displaystyle \sum _{K\in {\mathcal{T}}_h}\parallel u-{\mathbb{Q}}_h{\left(u\right)\parallel }_K^2+h_K^2{\parallel \nabla (u-{\mathbb{Q}}_h\left(u\right))\parallel }_K^2}& \le \hfill & {\displaystyle h^{2(s+1)}{\parallel u\parallel }_{s+1}^2,}\hfill \\ \hfill {\displaystyle \sum _{K\in {\mathcal{T}}_h}{\parallel \nabla (u-{\mathbb{Q}}_h\left(u\right))\parallel }_K^2}& \le \hfill & {\displaystyle h^{2s}{\parallel u\parallel }_{s+1}^2.}\hfill \end{array}\end{array}$$

(22)

Next, introduce the linear form $\phi (·)$ defined by

$$\begin{array}{ccc}\hfill \phi \left(v\right)& :=& {(\sigma ({\mathbb{Q}}_h\left(u\right)-u),v^0)}_{\Omega }+\sum _{K\in {\mathcal{T}}_h}{⟨(v^b-v^0)\wedge n,{\mathbf{Q}}_h\left(\mathrm{curl}u\right)-\mathrm{curl}u⟩}_{\partial K}\hfill \\ & & -s_K({\mathbb{Q}}_h\left(u\right),v)+\sum _{K\in {\mathcal{T}}_h}{⟨(v^b-v^0)·n,p-Q_h\left(p\right)⟩}_{\partial K}.\hfill \end{array}$$

Therefore, the error equations in Lemma 3 may be written as the form

$$\begin{array}{ccc}\hfill A_s(e_h,v)+B(v,{\epsilon }_h)& =& \phi \left(v\right),\hfill \\ \hfill B(e_h,\psi )& =& 0.\hfill \end{array}$$

Since the two bilinear forms $A_s$ and B satisfy all conditions of the well-known Babuška and Brezzi theory, it follows that $\left|\left|\left|e_h\right|\right|\right|+\parallel {\epsilon }_h{\parallel \le C\parallel \phi \parallel }_{{{\mathcal{V}}_h^0}^{\prime }}.$ Thus, it is sufficient to bound ${\parallel \phi \parallel }_{{{\mathcal{V}}_h^0}^{\prime }}$. For an estimation of the term ${(\sigma ({\mathbb{Q}}_h\left(u\right)-u),v^0)}_{\Omega }$, since $\sigma$ is bounded, ${\parallel ·\parallel }_{s+1}\le {\parallel ·\parallel }_{s+2}$, the first inequality in (22) and the definition of $\left|\left|\left|·\right|\right|\right|$ obtain

$$\begin{array}{ccc}\hfill \sum _{K\in {\mathcal{T}}_h}{(\sigma ({\mathbb{Q}}_h\left(u\right)-u),v^0)}_K& \le & C\sum _{K\in {\mathcal{T}}_h}\left|{({\mathbb{Q}}_h\left(u\right)-u,v^0)}_K\right|\hfill \\ & \le & C\sum _{K\in {\mathcal{T}}_h}\left|\right|{\mathbb{Q}}_h\left(u\right)-u\left|\right|\left|\right|v^0\left|\right|\hfill \\ & \le & C{h}^{s+1}{\parallel u\parallel }_{s+1}\left|\right|v^0\left|\right|\hfill \\ & \le & C{h}^{s+1}{\parallel u\parallel }_{s+1}\left|\left|\left|v\right|\right|\right|\le C{h}^{s+1}{\parallel u\parallel }_{s+2}\left|\left|\left|v\right|\right|\right|.\hfill \end{array}$$

From the definition of the projection operators ${\mathbb{Q}}_0$, ${\mathbb{Q}}_b$, Cauchy Schwarz inequality, (10), (13), and (22), one can arrive to (see [13] for similar details)

$$\begin{array}{ccc}& & \delta \sum _{K\in {\mathcal{T}}_h}\left|h_K^{-1}{⟨({\mathbb{Q}}_b\left(u\right)-{\mathbb{Q}}_0\left(u\right))\wedge n,(v^0-v^b)\wedge n⟩}_{\partial K}\right|\hfill \\ & =& \delta \sum _{K\in {\mathcal{T}}_h}\left|h_K^{-1}{⟨(u-{\mathbb{Q}}_0\left(u\right))\wedge n-(u-{\mathbb{Q}}_b\left(u\right))\wedge n,(v^0-v^b)\wedge n⟩}_{\partial K}\right|\hfill \\ & \le & C\left|\left|\left|v\right|\right|\right|{\left(\sum _{K\in {\mathcal{T}}_h}\left(h_K^{-2}\parallel {\mathbb{Q}}_0{\left(u\right)-u\parallel }_T^2+{\parallel \nabla ({\mathbb{Q}}_0\left(u\right)-u)\parallel }_T^2\right)\right)}^{{\displaystyle \frac{1}{2}}}\le C{h}^{s+1}\left|\left|\left|v\right|\right|\right|{\parallel u\parallel }_{s+2}.\hfill \end{array}$$

and

$$\begin{array}{c}\hfill \delta \sum _{K\in {\mathcal{T}}_h}|h_K^{-1}{⟨({\mathbb{Q}}_b\left(u\right)-{\mathbb{Q}}_0\left(u\right))·n,(v^0-v^b)·n⟩}_{\partial K}|\le C{h}^{s+1}\left|\left|\left|v\right|\right|\right|{\parallel u\parallel }_{s+2}\end{array}$$

and from the two last estimations, we deduce that

$$\begin{array}{c}\hfill |s_K({\mathbb{Q}}_h\left(u\right),v)|\le C{h}^{s+1}\left|\left|\left|v\right|\right|\right|{\parallel u\parallel }_{s+2}.\end{array}$$

Now, let us find an estimation of ${\sum }_{K\in {\mathcal{T}}_h}{⟨(v^b-v^0)\wedge n,{\mathbf{Q}}_h\left(\mathrm{curl}u\right)-\mathrm{curl}u⟩}_{\partial K}$; for simplicity of writing, we denote this term by $\mathcal{R}(u,v)$. Apply the Cauchy Schwarz inequality, then

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill \mathcal{R}(u,v)& :=\hfill & {\displaystyle |\sum _{K\in {\mathcal{T}}_h}{⟨v^b\wedge n-v^0\wedge n,{\mathbf{Q}}_h\left(\mathrm{curl}u\right)-\mathrm{curl}u⟩}_{\partial K}|}\hfill \\ & \le \hfill & {\displaystyle \sum _{K\in {\mathcal{T}}_h}\left|{⟨(v^b\wedge n-v^0\wedge n),{\mathbf{Q}}_h\left(\mathrm{curl}u\right)-\mathrm{curl}u⟩}_{\partial K}\right|}\hfill \\ & \le \hfill & {\displaystyle \sum _{K\in {\mathcal{T}}_h}{h}_K^{-\frac{1}{2}}\parallel v^b\wedge n-v^0{\wedge n\parallel }_{0,\partial K}{h}_K^{\frac{1}{2}}{\parallel {\mathbf{Q}}_h\left(\mathrm{curl}u\right)-\mathrm{curl}u\parallel }_{\partial K}.}\hfill \end{array}\end{array}$$

Using the definition of $\left|\left|\left|·\right|\right|\right|$, the trace inequality (13), we can arrive at

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill \mathcal{R}(u,v)& \le \hfill & {\displaystyle {\left(\sum _{K\in {\mathcal{T}}_h}{h}_K^{-1}{\parallel (v^b-v^0)\wedge n\parallel }_{0,\partial K}^2\right)}^{\frac{1}{2}}{\left(\sum _{K\in {\mathcal{T}}_h}{h}_K{\parallel {\mathbf{Q}}_h\left(\mathrm{curl}u\right)-\mathrm{curl}u\parallel }_{\partial K}^2\right)}^{\frac{1}{2}}}\hfill \\ & \le \hfill & {\displaystyle \left|\left|\left|v\right|\right|\right|{\left(\sum _{K\in {\mathcal{T}}_h}{h}_K{\parallel {\mathbf{Q}}_h\left(\mathrm{curl}u\right)-\mathrm{curl}u\parallel }_{\partial K}^2\right)}^{\frac{1}{2}}}\hfill \\ & \le \hfill & {\displaystyle \left|\left|\left|v\right|\right|\right|{\left(\sum _{K\in {\mathcal{T}}_h}(\parallel {\mathbf{Q}}_h{\left(\mathrm{curl}u\right)-\mathrm{curl}u\parallel }_K^2+h_K^2{\parallel \nabla ({\mathbf{Q}}_h\left(\mathrm{curl}u\right)-\mathrm{curl}u)\parallel }_K^2\right)}^{\frac{1}{2}}.}\hfill \end{array}\end{array}$$

Next, we use (22) for $\mathrm{curl}u$ and write

$$\begin{array}{c}\hfill {\displaystyle |\sum _{K\in {\mathcal{T}}_h}{⟨(v^b-v^0)\wedge n,{\mathbf{Q}}_h\left(\mathrm{curl}u\right)-\mathrm{curl}u⟩}_{\partial K}|\le h^{s+1}{\parallel \mathrm{curl}u\parallel }_{s+1}\left|\left|\left|v\right|\right|\right|\le h^{s+1}{\parallel u\parallel }_{s+2}\left|\left|\left|v\right|\right|\right|.}\end{array}$$

The last technique may be applied for finding an estimation of the term ${\sum }_{K\in {\mathcal{T}}_h}{⟨v^b·n-v^0·n,p-Q_h\left(p\right)⟩}_{\partial K}$ without difficulty and we obtain

$$\begin{array}{c}\hfill |\sum _{K\in {\mathcal{T}}_h}{⟨v^b·n-v^0·n,p-Q_h\left(p\right)⟩}_{\partial K}|\le C{h}^{s+1}{\parallel p\parallel }_{s+1}\left|\left|\left|v\right|\right|\right|.\end{array}$$

From the previous estimations and the general saddle point problem of Babuška–Brezzi theory, the first inequality in Theorem 1 follows immediately and for the $L^2$ norm estimation, we can apply the same technique presented in [9,11] for the Laplacien operator. □

5. Numerical Tests

Now, we test numerically the model partial differential Equation (1) in a two dimensional space $\Omega$; our results were obtained using Matlab software. Here, $\Omega$ is considered as $\left\{\right(x,y\left)\right|0\le x,y\le 1\}$ partitioned into a uniform triangulation with equal size $h_i={\displaystyle \frac{1}{2^i}}$, $i=1,2,\cdots ,7$. The parameters $\epsilon$ and $\mu$ are assumed to be constants and to be equal to one, so $\epsilon \left(x\right)\equiv 1$ and $\mu \left(x\right)\equiv 1$. In the principal bilinear form A, a multiplicative coefficient parameter $\delta$ appears. This parameter is chosen to be equal to one, so $\delta =1$ in all of the following examples. Note that for any choice of $\delta >0$ that is not very large, the matrix associated with A is well conditioned and non-singular. For the cases where $\sigma =0$ and $\sigma \ne 0$ are studied separately, we present two examples for the case where $\sigma$ is null in $\Omega$ and two examples for the case $\sigma \ne 0$. The solutions $u^0$, $u^b$ are discritized with the $P_0$ piecewise polynomials on every triangle and edge, respectively. p is approximated with the $P_1$ piecewise polynomial functions on every triangle.

5.1. Example 1

In this example, we take $\sigma =0$ and a source J so that the true solution $(u,p)$ is $p(x,y)=(x^2-x)(y^2-y)exp\left(x\right)exp\left(y\right)$ and $u(x,y)={((y^2-y)exp\left(y\right),(x^2-x)exp\left(x\right))}^T.$ We show the results in Table 1.

Table 1. Numerical values for example 1.

h	$\left|\left|\left|{\mathit{e}}_{\mathit{h}}\right|\right|\right|$	Rate	$\parallel {\mathit{ϵ}}_{\mathit{h}}{\parallel }_{1,\mathit{h}}$	Rate	$\parallel {\mathit{u}}_{\mathit{h}}^0-{\mathbb{Q}}_{\mathit{h}}\left({\mathit{u}}^0\right)\parallel$	Rate
$2^{-1}$	1.3424 $\times {10}^0$	-	4.1287 $\times {10}^{-2}$	-	3.9425 $\times {10}^{-1}$	-
$2^{-2}$	6.8674 $\times {10}^{-1}$	9.6694 $\times {10}^{-1}$	1.1179 $\times {10}^{-2}$	1.8849	7.2533 $\times {10}^{-2}$	2.4424
$2^{-3}$	3.6357 $\times {10}^{-1}$	9.1754 $\times {10}^{-1}$	2.9763 $\times {10}^{-3}$	1.9092	1.8247 $\times {10}^{-2}$	1.9910
$2^{-4}$	1.8453 $\times {10}^{-1}$	9.7834 $\times {10}^{-1}$	7.5591 $\times {10}^{-4}$	1.9772	4.5773 $\times {10}^{-3}$	1.9951
$2^{-5}$	9.2617 $\times {10}^{-2}$	9.9453 $\times {10}^{-1}$	1.8972 $\times {10}^{-4}$	1.9943	1.1454 $\times {10}^{-3}$	1.9986
$2^{-6}$	4.6353 $\times {10}^{-2}$	9.9863 $\times {10}^{-1}$	4.7477 $\times {10}^{-5}$	1.9986	2.8642 $\times {10}^{-4}$	1.9996
$2^{-7}$	2.3182 $\times {10}^{-2}$	9.9966 $\times {10}^{-1}$	1.1872 $\times {10}^{-5}$	1.9996	7.1610 $\times {10}^{-5}$	1.9999

5.2. Example 2

Also here $\sigma$ is taken to be equal to zero, where the solution $(u,p)$ of (1) is $p(x,y)=sin\left((x^2-x)(y^2-y)\right)$ and $u(x,y)={(sin(y^2-y),sin(x^2-x))}^T.$ We show the values in Table 2.

Table 2. Numerical values for example 2.

h	$\left|\left|\left|{\mathit{e}}_{\mathit{h}}\right|\right|\right|$	Rate	$\parallel {\mathit{ϵ}}_{\mathit{h}}{\parallel }_{1,\mathit{h}}$	Rate	$\parallel {\mathit{u}}_{\mathit{h}}^0-{\mathbb{Q}}_{\mathit{h}}\left({\mathit{u}}^0\right)\parallel$	Rate
$2^{-1}$	5.4190 $\times {10}^{-1}$	-	2.5564 $\times {10}^{-2}$	-	1.2439 $\times {10}^{-1}$	-
$2^{-2}$	3.6407 $\times {10}^{-1}$	5.7382 $\times {10}^{-1}$	4.2902 $\times {10}^{-3}$	2.5750	3.3600 $\times {10}^{-2}$	1.8883
$2^{-3}$	1.9432 $\times {10}^{-1}$	9.0581 $\times {10}^{-1}$	1.0449 $\times {10}^{-3}$	2.0376	8.5859 $\times {10}^{-3}$	1.9684
$2^{-4}$	9.8704 $\times {10}^{-2}$	9.7723 $\times {10}^{-1}$	2.6050 $\times {10}^{-4}$	2.0041	2.1584 $\times {10}^{-3}$	1.9920
$2^{-5}$	4.9546 $\times {10}^{-2}$	9.9435 $\times {10}^{-1}$	6.5090 $\times {10}^{-5}$	2.0008	5.4036 $\times {10}^{-4}$	1.9980
$2^{-6}$	2.4797 $\times {10}^{-2}$	9.9859 $\times {10}^{-1}$	1.6271 $\times {10}^{-5}$	2.0002	1.3514 $\times {10}^{-4}$	1.9995
$2^{-7}$	1.2402 $\times {10}^{-2}$	9.9965 $\times {10}^{-1}$	4.0675 $\times {10}^{-5}$	2.0000	3.3787 $\times {10}^{-5}$	1.9999

5.3. Example 3

Here, we choose $\sigma =1$ and a source J so that the true solution $(u,p)$ is $p(x,y)=(y^2-y)(x^2-x)e^y{e}^x$ and $u(x,y)={((y^2-y)e^y,(x^2-x)e^x)}^T.$ The results of convergence for this example are listed in Table 3.

Table 3. Numerical values for example 3.

h	$\left|\left|\left|{\mathit{e}}_{\mathit{h}}\right|\right|\right|$	Rate	$\parallel {\mathit{ϵ}}_{\mathit{h}}{\parallel }_{1,\mathit{h}}$	Rate	$\parallel {\mathit{u}}_{\mathit{h}}^0-{\mathbb{Q}}_{\mathit{h}}\left({\mathit{u}}^0\right)\parallel$	Rate
$2^{-1}$	2.5110 $\times {10}^0$	-	4.1730 $\times {10}^{-2}$	-	8.4684 $\times {10}^{-1}$	-
$2^{-2}$	7.1255 $\times {10}^{-1}$	1.8172 $\times {10}^0$	1.1004 $\times {10}^{-2}$	1.9231	7.9787 $\times {10}^{-2}$	3.4079
$2^{-3}$	3.7473 $\times {10}^{-1}$	9.2714 $\times {10}^{-1}$	2.9644 $\times {10}^{-3}$	1.8921	1.9892 $\times {10}^{-2}$	2.0040
$2^{-4}$	1.9001 $\times {10}^{-1}$	9.7981 $\times {10}^{-1}$	7.5516 $\times {10}^{-4}$	1.9729	4.9849 $\times {10}^{-3}$	1.9965
$2^{-5}$	9.5343 $\times {10}^{-2}$	9.9485 $\times {10}^{-1}$	1.8967 $\times {10}^{-4}$	1.9933	1.2472 $\times {10}^{-3}$	1.9989
$2^{-6}$	4.7714 $\times {10}^{-2}$	9.9871 $\times {10}^{-1}$	4.7474 $\times {10}^{-5}$	1.9983	3.1186 $\times {10}^{-4}$	1.9997
$2^{-7}$	2.3862 $\times {10}^{-2}$	9.9968 $\times {10}^{-1}$	1.1872 $\times {10}^{-5}$	1.9996	7.7969 $\times {10}^{-5}$	1.9999

5.4. Example 4

Here $\sigma$ is taken as $\sigma =1$ and the manufactured solution $(u,p)$ of (1) is chosen as $p(x,y)=sin\left((y^2-y)(x^2-x)\right)$ and $u(x,y)={(sin(y^2-y),sin(x^2-x))}^T$. The errors are shown in Table 4.

Table 4. Numerical values for example 4.

h	$\left|\left|\left|{\mathit{e}}_{\mathit{h}}\right|\right|\right|$	Rate	$\parallel {\mathit{ϵ}}_{\mathit{h}}{\parallel }_{1,\mathit{h}}$	Rate	$\parallel {\mathit{u}}_{\mathit{h}}^0-{\mathbb{Q}}_{\mathit{h}}\left({\mathit{u}}^0\right)\parallel$	Rate
$2^{-1}$	8.8187 $\times {10}^{-1}$	-	2.6724 $\times {10}^{-2}$	-	2.0271 $\times {10}^{-1}$	-
$2^{-2}$	5.5315 $\times {10}^{-1}$	6.7289 $\times {10}^{-1}$	4.2981 $\times {10}^{-3}$	2.6363	4.9930 $\times {10}^{-2}$	2.0214
$2^{-3}$	2.9250 $\times {10}^{-1}$	9.1922 $\times {10}^{-1}$	1.0450 $\times {10}^{-3}$	2.0402	1.2738 $\times {10}^{-2}$	1.9707
$2^{-4}$	1.4831 $\times {10}^{-1}$	9.7987 $\times {10}^{-1}$	2.6050 $\times {10}^{-4}$	2.0042	3.2020 $\times {10}^{-3}$	1.9921
$2^{-5}$	7.4412 $\times {10}^{-2}$	9.9497 $\times {10}^{-1}$	6.5090 $\times {10}^{-5}$	2.0008	8.0160 $\times {10}^{-4}$	1.9980
$2^{-6}$	3.7238 $\times {10}^{-2}$	9.9874 $\times {10}^{-1}$	1.6271 $\times {10}^{-5}$	2.0002	2.0047 $\times {10}^{-4}$	1.9995
$2^{-7}$	1.8623 $\times {10}^{-2}$	9.9969 $\times {10}^{-1}$	4.0675 $\times {10}^{-5}$	2.0000	5.0122 $\times {10}^{-5}$	1.9999

We remark that in all examples, the numerical results show the convergence of $\left|\left|\left|e_h\right|\right|\right|$ according to the rate $O\left(h\right)$, $\parallel u_h-{\mathbb{Q}}_h\left(u\right)\parallel$ and $\parallel {ϵ}_h{\parallel }_h$ according to the rate $O\left(h^2\right)$, and they confirm the theoretical results proven in this paper.

6. Conclusions and Remarks

In this work, we proposed and rigorously analyzed a novel numerical scheme founded on the weak Galerkin mixed finite element method for the approximation of the electrostatic field, formulated from the modified Maxwell equations through the introduction of a Lagrange multiplier. We established the well-posedness of the method and derived optimal convergence rates, which were subsequently validated by numerical experiments. These results are encouraging and suggest that the proposed framework can be successfully generalized to other classes of partial differential equations, including reaction–diffusion and Stokes systems.