Analysis of a Weak Galerkin Mixed Formulation for Modified Maxwell’s Equations

Abstract

In this paper, we are interested in studying a mixed formulation of weak Galerkin type to approach the electric field and a Lagrange multiplier, which are solutions of a problem deriving from Maxwell’s equations. Our numerical scheme is formed with stable finite elements constructed of usual polynomials of degree k for the electric field and of degree $k+1$ for the Lagrange multiplier; its consistency and well-posedness are shown. Some optimal error estimates are proven and tested numerically in a bounded subdomain of ${\mathbb{R}}^2$.

1. Introduction

Maxwell’s equations present the foundation of classical electromagnetism; they describe how electric and magnetic fields propagate and interact with matter. Numerical methods for solving these equations are essential in many applications, from antenna design to electromagnetic field analysis in various materials. The numerical study of Maxwell’s equations, alone or coupled with other partial differential equations, occupies a significant part of the papers in the literature. In [1], a fully discrete finite element projection method for magnetohydrodynamic equations is presented and analyzed. In [2], the authors analyze a temporally second-order accurate numerical scheme for the Cahn–Hilliard magnetohydrodynamics system of equations. This scheme was based on a modified Crank–Nicolson-type approximation for the time discretization and a mixed finite element method for the spatial discretization. We refer, for example, to [3,4] for the numerical resolution of Maxwell’s equations using the discontinuous Galerkin method and to [5,6] for the study of Maxwell’s equations using weak Galerkin techniques. The motivation for the use of the weak Galerkin formulation lies in its ability to handle discontinuities and efficiency in solving complex problems in irregular domains with adaptive meshes. Its capability to work with non-conforming meshes, adapt to evolving geometries, and model weakly discontinuous solutions makes it a powerful tool for a wide range of applications, including fluid dynamics, structural analysis, and multi-physics problems. The method provides a robust and numerically stable framework, especially when dealing with singularities, complex boundary conditions, and high-order accuracy requirements. Weak Galerkin methods present a class of numerical techniques designed to solve partial differential equations by combining concepts from the habitual Galerkin methods with weak formulations. These methods have been used by several researchers for approximating solutions of PDEs. We refer to [7,8,9] for linear parabolic problems, ref. [10] for Helmholtz equations with large wave numbers, refs. [11,12] for elliptic interface problems, ref. [13] for an analysis of a mixed weak Galerkin scheme for the second-order elliptic problem, and [14,15,16,17] for other kinds of partial differential problems.

Let $\Omega$ be a bounded domain of ${\mathbb{R}}^2$ or ${\mathbb{R}}^3$ and $\Gamma$ its boundary. In this work, we present and study a weak Galerkin mixed formulation for the following partial differential equations which is derived from Maxwell’s equations as follows: Find u satisfying

$$\begin{array}{cc}\hfill \nabla \wedge ({\mu }^{-1}\nabla \wedge \mathbf{u})=& J\mathrm{in}\Omega ,\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \nabla ·\left(\epsilon \mathbf{u}\right)=& 0\mathrm{in}\Omega ,\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill n\wedge \mathbf{u}=& 0\mathrm{on}\Gamma .\hfill \end{array}$$

(3)

It is well known from the literature that we can introduce a new variable, p, called a Lagrange multiplier, and then we can use mixed formulation to find u and p. Let us introduce a new function p, which plays the role of a Lagrange multiplier, and consider the partial differential equations: Find u and p such that

$$\begin{array}{cc}\hfill \nabla \wedge ({\mu }^{-1}\nabla \wedge \mathbf{u})-\epsilon \nabla p=& J\mathrm{in}\Omega ,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \nabla ·\left(\epsilon \mathbf{u}\right)-\alpha p=& 0\mathrm{in}\Omega ,\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill \nabla ·\left(\epsilon \mathbf{u}\right)=& 0\mathrm{in}\Omega ,\hfill \end{array}$$

(6)

together with the boundary conditions

$$\begin{array}{cc}\hfill n\wedge \mathbf{u}=& 0\mathrm{on}\Gamma ,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill p=& 0\mathrm{on}\Gamma .\hfill \end{array}$$

(8)

It is clear from Equations (5) and (6) that the introduced variable p must be zero and the new system is the same as the first system without the variable p. Also, when we select $\alpha =0$ in (5), then the term $\alpha p$ vanishes, making the Equations in (5) and (6) identical. In this paper, Equation (6) is introduced in our formulation as a penalization term and is added to Equation (4) in the mixed weak Galerkin formulation given later. Equation (6) is used to determine two bilinear forms, B and C, which means applying a weak Galerkin mixed formulation as a saddle typical perturbed problem. In the literature, we find that the electric field $E(t,x)$ is the real part of $\mathbf{u}\left(x\right)exp\left(i\omega t\right)$, with a non-null frequency w. p is the Lagrange multiplier introduced to relax the divergence of the electrostatic field E. $\mu$ and $\epsilon$ are, respectively, the magnetic permeability and electric permittivity of the media. We suppose that these parameters are strictly positive, sufficiently smooth, bounded below and above, and that $\alpha$ is a bounded positive function in $\Omega$. The notations and functional spaces used in this paper are well detailed in [3]. Assume that ${\mathcal{T}}_h$ is a shape-regular subdivision of $\Omega$ which consists of triangles if $\Omega \subset {\mathbb{R}}^2$ or of tetrahedra if $\Omega \subset {\mathbb{R}}^3$. We use ${\mathcal{E}}_h^I$ to denote the set of all interior boundary elements of ${\mathcal{T}}_h$, ${\mathcal{E}}_h^D$ the set of all exterior boundary elements of ${\mathcal{T}}_h$, and ${\mathcal{E}}_h$ the set of all boundary elements of ${\mathcal{T}}_h$. Given $s>0$ and $d\in \{1,2,3\}$, we introduce the broken Sobolev spaces:

$$H^s{\left({\mathcal{T}}_h\right)}^d:=\left\{\mathbf{v}\in L^2{(\Omega )}^d:{\mathbf{v}}_{|K}\in H^s{\left(K\right)}^d\mathrm{for}\mathrm{all}K\in {\mathcal{T}}_h\right\}.$$

We introduce the finite element spaces needed in this paper as

$$\begin{array}{cc}\hfill {\mathcal{V}}_h:=& \left\{\mathbf{v}=\{{\mathbf{v}}^0,{\mathbf{v}}^b\}:{\mathbf{v}}^0\in {\left[P_k\left(T\right)\right]}^d,{\mathbf{v}}^b\in {\left[P_k\left(e\right)\right]}^d,e\in {\mathcal{E}}_h,T\in {\mathcal{T}}_h\right\},\hfill \\ \hfill {\mathcal{W}}_h:=& \left\{\psi \in L^2(\Omega ):{\psi }_{|T}\in P_{k+1}\left(T\right),T\in {\mathcal{T}}_h\right\}.\hfill \end{array}$$

and their subspaces as

$$\begin{array}{cc}\hfill {\mathcal{V}}_h^0:=& \left\{\mathbf{v}\in {\mathcal{V}}_h:{\mathbf{v}}^b\wedge n=0\mathrm{on}\partial \Omega \right\},\hfill \\ \hfill {\mathcal{W}}_h^0:=& \left\{\psi \in {\mathcal{W}}_h:\psi =0\mathrm{on}\partial \Omega \right\}.\hfill \end{array}$$

Now, for any element in ${\mathcal{T}}_h$ and $v\in {\mathcal{T}}_h$, we recall the definition of the weak divergence operator ${\nabla }_w·\mathbf{v}\in P_k\left(T\right)$ as the unique polynomial such that

$${({\nabla }_w·\mathbf{v},\varphi )}_T=-{({\mathbf{v}}^0,\nabla \varphi )}_T+{⟨{\mathbf{v}}^b·n,\varphi ⟩}_{\partial T}\mathrm{for}\mathrm{all}\varphi \in P_k\left(T\right)$$

(9)

and the weak curl ${\nabla }_w\wedge \mathbf{v}\in {\left[P_k\left(T\right)\right]}^d$ is defined as the unique polynomial such that

$${({\nabla }_w\wedge \mathbf{v},\mathbf{w})}_T={({\mathbf{v}}^0,\nabla \wedge \mathbf{w})}_T-{⟨{\mathbf{v}}^b\wedge n,\mathbf{w}⟩}_{\partial T}\mathrm{for}\mathrm{all}\mathbf{w}\in {\left[P_k\left(T\right)\right]}^d.$$

(10)

2. Weak Galerkin Formulation

In order to obtain a consistent numerical formulation for the partial differential equation system, a sufficient enforcement of the components $u^b$ and $u^0$ is necessary; therefore, we introduce the bilinear form

$$\begin{array}{cc}\hfill a_{TN}({\mathbf{u}}_h,{\mathbf{v}}_h):=& r\sum _{\partial T\in {\mathcal{E}}_h^I}{h}_T^{-1}{⟨(\epsilon {\mathbf{u}}_h^0-\epsilon {\mathbf{u}}_h^b)·n,(\epsilon {\mathbf{v}}_h^0-\epsilon {\mathbf{v}}_h^b)·n⟩}_{\partial T}\hfill \\ & +r\sum _{\partial T\in {\mathcal{E}}_h}{h}_T^{-1}{⟨({\mathbf{u}}_h^0-{\mathbf{u}}_h^b)\wedge n,({\mathbf{v}}_h^0-{\mathbf{v}}_h^b)\wedge n⟩}_{\partial T}\hfill \end{array}$$

which introduces some communications between the tangential and normal components of ${\mathbf{u}}^b$ and ${\mathbf{u}}^0$. Next, a weak Galerkin mixed formulation for an approximate solution of (4)–(6) together with conditions (7) and (8) is as follows: Find $p_h\in {\mathcal{W}}_h^0$ and ${\mathbf{u}}_h=\{{\mathbf{u}}_h^0,{\mathbf{u}}_h^b\}\in {\mathcal{V}}_h^0$, such that

$$\begin{array}{cc}\hfill A({\mathbf{u}}_h,{\mathbf{v}}_h)+B({\mathbf{v}}_h,p_h)=& \sum _{T\in {\mathcal{T}}_h}(J,{\mathbf{v}}_h^0)\mathrm{for}\mathrm{all}{\mathbf{v}}_h\in {\mathcal{V}}_h^0,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill B({\mathbf{u}}_h,{\psi }_h)-C(p_h,{\psi }_h)=& 0\mathrm{for}\mathrm{all}{\psi }_h\in {\mathcal{W}}_h^0\hfill \end{array}$$

(12)

where we denote

$$\begin{array}{cc}\hfill a({\mathbf{u}}_h,{\mathbf{v}}_h):=& \sum _{T\in {\mathcal{T}}_h}{({\mu }^{-1}{\nabla }_w\wedge {\mathbf{u}}_h,{\nabla }_w\wedge {\mathbf{v}}_h)}_T+{({\nabla }_w·\epsilon {\mathbf{u}}_h,{\nabla }_w·\epsilon {\mathbf{v}}_h)}_T,\hfill \\ \hfill B({\mathbf{v}}_h,p_h):=& \sum _{T\in {\mathcal{T}}_h}{(p_h,{\nabla }_w·\epsilon v_h)}_T,\hfill \\ \hfill C(p_h,{\psi }_h):=& \sum _{T\in {\mathcal{T}}_h}{(\alpha p_h,{\psi }_h)}_T,\hfill \\ \hfill A({\mathbf{u}}_h,{\mathbf{v}}_h):=& a({\mathbf{u}}_h,{\mathbf{v}}_h)+a_{TN}({\mathbf{u}}_h,{\mathbf{v}}_h).\hfill \end{array}$$

In the following theorem, we study the consistency of the numerical scheme (11) and (12); we present the following.

Theorem 1.

The mixed WG numerical scheme (11) and (12) has just one solution $({\mathbf{u}}_h,p_h)$ in ${\mathcal{V}}_h^0\times {\mathcal{W}}_h^0$.

Proof. 

Suppose that $J=0$ in (11) and (12). Then, we must show that ${\mathbf{u}}_h=p_h=0$ everywhere in $\Omega$. Let $(\mathbf{v},\psi )=({\mathbf{v}}_h,p_h)$ in (11) and (12) and subtract the second equation from the first equation in (11) and (12). We obtain

$$A({\mathbf{u}}_h,{\mathbf{u}}_h)+C(p_h,p_h)=0.$$

From the positivity of $C(p_h,p_h)$, we deduce that $A({\mathbf{u}}_h,{\mathbf{u}}_h)=0$ and this means that ${\nabla }_w\wedge {\mathbf{u}}_h=0$, ${\nabla }_w·\epsilon {\mathbf{u}}_h=0$ for any $T\in {\mathcal{T}}_h$, $\epsilon {\mathbf{u}}^0·n=\epsilon {\mathbf{u}}^b·n$ and ${\mathbf{u}}^0\wedge n={\mathbf{u}}^b\wedge n$, for any edge e in ${\mathcal{E}}_h$. Using the definition of the weak curl (10), the fact that ${\nabla }_w\wedge {\mathbf{u}}_h=0$, one has

$$\begin{array}{cc}\hfill 0={({\nabla }_w\wedge \mathbf{u},\mathbf{w})}_T={({\mathbf{u}}^0,\nabla \wedge \mathbf{w})}_T-{⟨{\mathbf{u}}^b\wedge n,\mathbf{w}⟩}_{\partial T}=& {(\nabla \wedge {\mathbf{u}}^0,\mathbf{w})}_T-{⟨({\mathbf{u}}^0-{\mathbf{u}}^b)\wedge n,\mathbf{w}⟩}_{\partial T}\hfill \\ \hfill =& {(\nabla \wedge {\mathbf{u}}^0,\mathbf{w})}_T\hfill \end{array}$$

for any $\mathbf{w}$ in $P_k{\left(T\right)}^d$, which gives $\nabla \wedge {\mathbf{u}}^0=0$ for every element T in ${\mathcal{T}}_h$. Since ${\mathbf{u}}^0\wedge n={\mathbf{u}}^b\wedge n$ on every element of ${\mathcal{E}}_h$ and ${\mathbf{u}}^b\wedge n=0$ on the boundary of $\Omega$, one can arrive at ${\mathbf{u}}^0\in H_0(\nabla \times ,\Omega )$ with $\nabla \wedge {\mathbf{u}}^0$ null everywhere in $\Omega$. Similarly, the fact that ${\nabla }_w·\epsilon \mathbf{u}=0$ for each element of ${\mathcal{T}}_h$ and $\epsilon {\mathbf{u}}^b·n=\epsilon {\mathbf{u}}^0·n$ for every element of ${\mathcal{E}}_h$ implies that ${\mathbf{u}}^0\in H({\nabla }_{\epsilon }·,\Omega )$ with $\nabla ·\epsilon {\mathbf{u}}^0$ null everywhere in $\Omega$ which means that ${\mathbf{u}}^0=0$ in $\Omega$. Hence, ${\mathbf{u}}^b\wedge n=\epsilon {\mathbf{u}}^b·n=0$ and then ${\mathbf{u}}^b=0$ in ${\mathcal{T}}_h$. Now, the definition of the bilinear form B and the weak divergence operator (9) together with the first equation in (11) and (12) means that $p_h=0$, and this finish the proof. □

3. Error Estimations

In this paragraph, we define the local projection operators and denote by ${\mathbb{Q}}_0$ the projection from ${\left(L^2\left(T\right)\right)}^d$ to ${\left(P_k\left(T\right)\right)}^d$. The projection from ${\left(L^2\left(e\right)\right)}^d$ to ${\left(P_k\left(e\right)\right)}^d$ on each element $T\in {\mathcal{T}}_h$ and $e\in {\mathcal{E}}_h$ is denoted by ${\mathbb{Q}}_b$. For any $\mathbf{v}=\{{\mathbf{v}}^0,{\mathbf{v}}^b\}\in {\mathcal{V}}_h^0$, we denote by ${\mathbb{Q}}_h$ its $L^2$ projection, defined by ${\mathbb{Q}}_h:=\{{\mathbb{Q}}_0\left({\mathbf{v}}^0\right),{\mathbb{Q}}_b\left({\mathbf{v}}^b\right)\}.$ For a given $p\in {\mathcal{W}}_h^0$, we design, using $Q_h\left(p\right)$, its local projection from $L^2\left(T\right)$ onto $P_{k+1}\left(T\right)$. With the previous notations, we can obtain the following equations, which we need to establish the error equations that are primordial. For the establishment of some error estimations, we refer to [3] for a rigorous proof of these equations. Assume that $(\mathbf{u},p)$ is the true solution of (4)–(6); then, one has

$$\begin{array}{cc}\hfill {({\nabla }_w\wedge \mathbf{u},{\mathbb{Q}}_h\left(\phi \right))}_T=& {({\mathbf{u}}^0,\nabla \wedge \phi )}_T+{⟨({\mathbf{u}}^b-{\mathbf{u}}^0)\wedge n,\phi -{\mathbb{Q}}_h\left(\phi \right)⟩}_{\partial T}-{⟨{\mathbf{u}}^b\wedge n,\phi ⟩}_{\partial T},\hfill \\ \hfill {\nabla }_w·{\left({\mathbb{Q}}_h\left(\mathbf{u}\right)\right)}_T=& Q_h{(\nabla ·\mathbf{u})}_T,\hfill \\ \hfill {\nabla }_w\wedge {\left({\mathbb{Q}}_h\left(\mathbf{u}\right)\right)}_T=& {\mathbf{Q}}_h{(\nabla \wedge \mathbf{u})}_T.\hfill \end{array}$$

(13)

Now, we establish some error equations which we use to derive optimal error estimations for our WG mixed finite element formulation (11) and (12).

Error Equations

Assume that $\mathbf{u}$ and p are sufficiently smooth solutions of (4)–(6) and, for the sake of simplicity, assume that the parameters $\mu$ and $\epsilon$ are constant. The use of (13), the definition of weak curl (10), implies

$$\begin{array}{cc}\hfill {({\nabla }_w\wedge \left({\mathbb{Q}}_h\left(\mathbf{u}\right)\right),{\nabla }_w\wedge \mathbf{v})}_T=& {({\mathbf{Q}}_h(\nabla \wedge \mathbf{u}),{\nabla }_w\wedge \mathbf{v})}_T\hfill \\ \hfill =& {({\mathbf{v}}^0,\nabla \wedge {\mathbf{Q}}_h(\nabla \wedge \mathbf{u}))}_T-{⟨{\mathbf{v}}^b\wedge n,{\mathbf{Q}}_h(\nabla \wedge \mathbf{u})⟩}_{\partial T}.\hfill \end{array}$$

Therefore, we can obtain, after an integration by parts,

$$\begin{array}{cc}\hfill {({\nabla }_w\wedge \left({\mathbb{Q}}_h\left(\mathbf{u}\right)\right),{\nabla }_w\wedge \mathbf{v})}_T=& {(\nabla \wedge {\mathbf{v}}^0,{\mathbf{Q}}_h(\nabla \wedge \mathbf{u}))}_T+{⟨({\mathbf{v}}^0-{\mathbf{v}}^b)\wedge n,{\mathbf{Q}}_h(\nabla \wedge \mathbf{u})⟩}_{\partial T}\hfill \\ \hfill =& {(\nabla \wedge {\mathbf{v}}^0,\nabla \wedge \mathbf{u})}_T+{⟨({\mathbf{v}}^0-{\mathbf{v}}^b)\wedge n,{\mathbf{Q}}_h(\nabla \wedge \mathbf{u})⟩}_{\partial T}.\hfill \end{array}$$

(14)

Also, we use the definition of weak divergence (9) and write

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

It follows after an integration by parts and using the fact that ${\sum }_{T\in {\mathcal{T}}_h}{⟨{\mathbf{v}}^b·n,p⟩}_{\partial T}=0$, that

$$\begin{array}{cc}\hfill {({\nabla }_w·\mathbf{v},Q_h\left(p\right))}_{\Omega }=& \sum _{T\in {\mathcal{T}}_h}{(\nabla ·{\mathbf{v}}^0,Q_h\left(p\right))}_T-{⟨({\mathbf{v}}^0-{\mathbf{v}}^b)·n,Q_h\left(p\right)⟩}_{\partial T}\hfill \\ \hfill =& \sum _{T\in {\mathcal{T}}_h}{(\nabla ·{\mathbf{v}}^0,p)}_T-{⟨({\mathbf{v}}^0-{\mathbf{v}}^b)·n,Q_h\left(p\right)⟩}_{\partial T}\hfill \\ \hfill =& -\sum _{T\in {\mathcal{T}}_h}{({\mathbf{v}}^0,\nabla p)}_T+{⟨{\mathbf{v}}^0·n,p⟩}_{\partial T}\hfill \\ \hfill & -\sum _{T\in {\mathcal{T}}_h}{⟨({\mathbf{v}}^0-{\mathbf{v}}^b)·n,Q_h\left(p\right)⟩}_{\partial T}\hfill \\ \hfill =& -\sum _{T\in {\mathcal{T}}_h}{({\mathbf{v}}^0,\nabla p)}_T+{⟨({\mathbf{v}}^0-{\mathbf{v}}^b)·n,p⟩}_{\partial T}\hfill \\ \hfill & -\sum _{T\in {\mathcal{T}}_h}{⟨({\mathbf{v}}^0-{\mathbf{v}}^b)·n,Q_h\left(p\right)⟩}_{\partial T}\hfill \end{array}$$

which means that

$${({\nabla }_w·\mathbf{v},Q_h\left(p\right))}_{\Omega }=-{({\mathbf{v}}^0,\nabla p)}_{\Omega }+\sum _{T\in {\mathcal{T}}_h}{⟨({\mathbf{v}}^0-{\mathbf{v}}^b)·n,p-Q_h\left(p\right)⟩}_{\partial T}.$$

(15)

We multiply Equation (4) by ${\mathbf{v}}^0$ in $v=\{{\mathbf{v}}^0,{\mathbf{v}}^b\}\in V_h^0$ and obtain

$${(\nabla \wedge \nabla \wedge \mathbf{u},{\mathbf{v}}^0)}_{\Omega }-{(\nabla p,{\mathbf{v}}^0)}_{\Omega }={(J,{\mathbf{v}}^0)}_{\Omega }.$$

(16)

We integrate by parts the first term in the last equation and, since ${\sum }_{T\in {\mathcal{T}}_h}{⟨{\mathbf{v}}^b\wedge n,(\nabla \wedge \mathbf{u})⟩}_{\partial T}=0$, one has

$$\begin{array}{cc}\hfill {(\nabla \wedge \nabla \wedge \mathbf{u},{\mathbf{v}}^0)}_{\Omega }=& \sum _{T\in {\mathcal{T}}_h}{(\nabla \wedge \mathbf{u},\nabla \wedge {\mathbf{v}}^0)}_T\hfill \\ \hfill & +{⟨({\mathbf{v}}^0-{\mathbf{v}}^b)\wedge n,(\nabla \wedge \mathbf{u})⟩}_{\partial T}.\hfill \end{array}$$

The use of (14) together with this last equation means that

$$\begin{array}{cc}\hfill {(\nabla \wedge \nabla \wedge \mathbf{u},{\mathbf{v}}^0)}_{\Omega }=& {({\nabla }_w\wedge \left({\mathbb{Q}}_h\left(\mathbf{u}\right)\right),{\nabla }_w\wedge \mathbf{v})}_{\Omega }\hfill \\ \hfill & -\sum _{T\in {\mathcal{T}}_h}{⟨({\mathbf{v}}^0-{\mathbf{v}}^b)\wedge n,{\mathbf{Q}}_h(\nabla \wedge \mathbf{u})⟩}_{\partial T}\hfill \\ \hfill & +\sum _{T\in {\mathcal{T}}_h}{⟨({\mathbf{v}}^0-{\mathbf{v}}^b)\wedge n,(\nabla \wedge \mathbf{u})⟩}_{\partial T}\hfill \end{array}$$

which is equivalent to

$${(\nabla \wedge \nabla \wedge \mathbf{u},{\mathbf{v}}^0)}_{\Omega }={({\nabla }_w\wedge \left({\mathbb{Q}}_h\left(\mathbf{u}\right)\right),{\nabla }_w\wedge \mathbf{v})}_{\Omega }-\sum _{T\in {\mathcal{T}}_h}{⟨({\mathbf{v}}^0-{\mathbf{v}}^b)\wedge n,{\mathbf{Q}}_h(\nabla \wedge \mathbf{u})-\nabla \wedge \mathbf{u}⟩}_{\partial T}$$

(17)

Next, extracting the term $-({\mathbf{v}}^0,\nabla p)$ from (15), we substitute it together with (17) into (16) and obtain the first error equation as

$$\begin{array}{cc}\hfill {({\nabla }_w\wedge {\left({\mathbb{Q}}_h\left(\mathbf{u}\right)\right)}_{\Omega },{\nabla }_w\wedge \mathbf{v})}_{\Omega }+{({\nabla }_w·\mathbf{v},Q_h\left(p\right))}_{\Omega }=& {(J,{\mathbf{v}}^0)}_{\Omega }\hfill \\ & -\sum _{T\in {\mathcal{T}}_h}{⟨({\mathbf{v}}^b-{\mathbf{v}}^0)·n,p-Q_h\left(p\right)⟩}_{\partial T}\hfill \\ & -\sum _{T\in {\mathcal{T}}_h}{⟨({\mathbf{v}}^b-{\mathbf{v}}^0)\wedge n,{\mathbf{Q}}_h(\nabla \wedge \mathbf{u})-\nabla \wedge \mathbf{u}⟩}_{\partial T}.\hfill \end{array}$$

(18)

Now, testing the equation in (6) by a function ${\nabla }_w·\mathbf{v}$, we obtain

$$0={(\nabla ·\mathbf{u},{\nabla }_w·\mathbf{v})}_{\Omega }={(Q_h(\nabla ·\mathbf{u}),{\nabla }_w·\mathbf{v})}_{\Omega }={({\nabla }_w·\left({\mathbb{Q}}_h\left(\mathbf{u}\right)\right),{\nabla }_w·\mathbf{v})}_{\Omega }.$$

(19)

The summation of Equations (18) and (19) implies that

$$\begin{array}{cc}\hfill ({\nabla }_w\wedge \left({\mathbb{Q}}_h\left(\mathbf{u}\right)\right),& {\nabla }_w{\wedge \mathbf{v})}_{\Omega }+{({\nabla }_w·\left({\mathbb{Q}}_h\left(\mathbf{u}\right)\right),{\nabla }_w·\mathbf{v})}_{\Omega }+{({\nabla }_w·\mathbf{v},Q_h\left(p\right))}_{\Omega }\hfill \\ \hfill =& {(J,{\mathbf{v}}^0)}_{\Omega }+\sum _{T\in {\mathcal{T}}_h}{⟨({\mathbf{v}}^0-{\mathbf{v}}^b)·n,p-Q_h\left(p\right)⟩}_{\partial T}\hfill \\ & +\sum _{T\in {\mathcal{T}}_h}{⟨({\mathbf{v}}^0-{\mathbf{v}}^b)\wedge n,{\mathbf{Q}}_h(\nabla \wedge \mathbf{u})-\nabla \wedge \mathbf{u}⟩}_{\partial T}.\hfill \end{array}$$

(20)

Next, we present and prove our error equations in the following.

Lemma 1.

The errors ${\mathbf{e}}_h:={\mathbf{u}}_h-{\mathbb{Q}}_h\left(\mathbf{u}\right)$ and ${ϵ}_h:=p_h-Q_h\left(p\right)$ satisfy the equations

$$\begin{array}{cc}\hfill A({\mathbf{e}}_h,\mathbf{v})+B(\mathbf{v},{ϵ}_h)=& \sum _{T\in {\mathcal{T}}_h}{⟨({\mathbf{v}}^b-{\mathbf{v}}^0)\wedge n,{\mathbf{Q}}_h(\nabla \wedge \mathbf{u})-\nabla \wedge \mathbf{u}⟩}_{\partial T}\hfill \\ \hfill & +\sum _{T\in {\mathcal{T}}_h}{⟨({\mathbf{v}}^b-{\mathbf{v}}^0)·n,p-Q_h\left(p\right)⟩}_{\partial T}\hfill \\ \hfill & -a_{TN}({\mathbb{Q}}_h\left(\mathbf{u}\right),\mathbf{v})\hfill \\ \hfill B({\mathbf{e}}_h,\psi )-C({ϵ}_h,\psi )=& 0.\hfill \end{array}$$

Proof. 

If we add the term $a_{TN}({\mathbb{Q}}_h\left(\mathbf{u}\right),\mathbf{v})$ to the left and right sides of (20), we obtain

$$\begin{array}{cc}\hfill A({\mathbb{Q}}_h\left(\mathbf{u}\right),\mathbf{v})+B(\mathbf{v},Q_h\left(p\right))=& {(J,{\mathbf{v}}^0)}_{\Omega }+a_{TN}({\mathbb{Q}}_h\left(\mathbf{u}\right),\mathbf{v})\hfill \\ & +\sum _{T\in {\mathcal{T}}_h}{⟨({\mathbf{v}}^0-{\mathbf{v}}^b)\wedge n,{\mathbf{Q}}_h(\nabla \wedge \mathbf{u})-\nabla \wedge \mathbf{u}⟩}_{\partial T}\hfill \\ & +\sum _{T\in {\mathcal{T}}_h}{⟨({\mathbf{v}}^0-{\mathbf{v}}^b)·n,p-Q_h\left(p\right)⟩}_{\partial T}.\hfill \end{array}$$

(21)

Subtracting (21) from (11), we arrive at

$$\begin{array}{cc}\hfill A({\mathbf{e}}_h,\mathbf{v})+B(\mathbf{v},{ϵ}_h)=& -a_{TN}({\mathbb{Q}}_h\left(\mathbf{u}\right),\mathbf{v})\hfill \\ & +\sum _{T\in {\mathcal{T}}_h}{⟨({\mathbf{v}}^b-{\mathbf{v}}^0)\wedge n,{\mathbf{Q}}_h(\nabla \wedge \mathbf{u})-\nabla \wedge \mathbf{u}⟩}_{\partial T}\hfill \\ & +\sum _{T\in {\mathcal{T}}_h}{⟨({\mathbf{v}}^b-{\mathbf{v}}^0)·n,p-Q_h\left(p\right)⟩}_{\partial T}.\hfill \end{array}$$

Multiplying (5) by a function $\psi$, we obtain

$$\begin{array}{cc}\hfill 0={(\nabla ·\mathbf{u},\psi )}_{\Omega }-{(\alpha p,\psi )}_{\Omega }=& {(Q_h(\nabla ·\mathbf{u}),\psi )}_{\Omega }-{(Q_h\left(\alpha p\right),\psi )}_{\Omega }\hfill \\ \hfill =& {({\nabla }_w·\left({\mathbb{Q}}_h\left(\mathbf{u}\right)\right),\psi )}_{\Omega }-{(Q_h\left(\alpha p\right),\psi )}_{\Omega }\hfill \\ \hfill =& B({\mathbb{Q}}_h\left(\mathbf{u}\right),\psi )-C(Q_h\left(p\right),\psi )\hfill \end{array}$$

which means that

$$B({\mathbb{Q}}_h\left(\mathbf{u}\right),\psi )-C(Q_h\left(p\right),\psi )=0.$$

This last equation, subtracted from (12), gives

$$B({\mathbf{e}}_h,\psi )-C({ϵ}_h,\psi )=0.$$

□

Since our mixed WG scheme (11) and (12) is a typical saddle-point problem, we can apply the well-known Babuška–Brezzi theory [18,19] for studying its error analysis. For this reason, we are forced to study the properties of the bilinear forms introduced in our formulation. Let us begin by defining norms on the spaces considered in the numerical scheme. For a fixed element $\mathbf{u}$ in ${\mathcal{V}}_h^0$ and a fixed element $\psi$ in ${\mathcal{W}}_h^0$, we introduce the norms ${\left|\left|\left|\mathbf{u}\right|\right|\right|}^2:=A(\mathbf{u},\mathbf{u})$ and $\parallel \psi \parallel ={\parallel \psi \parallel }_{0,\Omega }$ as the classical $L^2$ norm of $\psi$. Using the Cauchy–Schwarz inequalities, we notice that the bilinear forms that appear in (11) and (12) are continuous as well as that the principal form A is coercive on V. It remains to demonstrate only that the bilinear form B satisfies a condition inf-sup; this is the aim of the following lemma, and for its proof, we refer to [3].

Lemma 2.

The bilinear form B introduced in (11) and (12) satisfies the following property: for any $\psi \in {\mathcal{W}}_h^0$, there exists $\mathbf{v}\in {\mathcal{V}}_h^0$ such that

$$B(\psi ,\mathbf{v})\ge \beta \left|\left|\left|\mathbf{v}\right|\right|\right|\parallel \psi \parallel$$

with a positive constant β.

Once the three bilinear forms are continuous, A is coercive and B satisfies a condition inf-sup, we can apply the Babuška–Brezzi theory to establish convergence estimates. However, a very good detail for the derivation of error estimates is established in [3]. We have the following convergence result.

Theorem 2.

Assume that $({\mathbf{u}}_h,p_h)\in {\mathcal{V}}_h^0\times {\mathcal{W}}_h^0$ is the numerical solution of (11) and (12), $(\mathbf{u},p)$ is the true solution of (4)–(6) satisfying (7) and (8), and suppose that $(\mathbf{u},p)$ is in $H^{k+2}(\Omega )\times H^{k+1}(\Omega )$ with $k\ge 0$; then, one has the convergence results

$$\left|\left|\left|{\mathbb{Q}}_h\left(\mathbf{u}\right)-{\mathbf{u}}_h\right|\right|\right|+\parallel Q_h\left(p\right)-p_h\parallel \le C{h}^{s+1}\left({\parallel \mathbf{u}\parallel }_{s+2}+{\parallel p\parallel }_{s+1}\right)$$

(22)

and

$$\parallel {\mathbb{Q}}_0\left(\mathbf{u}\right)-{\mathbf{u}}_0\parallel \le C{h}^{k+2}\left({\parallel \mathbf{u}\parallel }_{k+2}+{\parallel p\parallel }_{k+1}\right).$$

(23)

In the remainder of this paper, we are interested in numerically studying the weak Galerkin scheme; we study separately the cases where $\alpha =0$ and $\alpha \ne 0$.

4. Numerical Tests

In this part, we test the convergence of our numerical scheme with two examples, one example for the case where $\alpha =0$ and another example for the non-zero case $(\alpha \ne 0)$. For reasons of simplicity, we assume that the physical coefficients $\mu$ and $\epsilon$ are constant and equal to unity. The domain considered in these examples is the unit square $(0,1)\times (0,1)$ and the numerical solution $({\mathbf{u}}_h,p_h)$ is discretized with the minimum order $k=0$ on the space ${\mathcal{V}}_h^0\times {\mathcal{W}}_h^0$. We note that the numerical results obtained in the two examples demonstrate and confirm the theoretical convergence results (22) and (23).

4.1. Example 1

In this test, we assume that the exact solution of our PDE system together with its boundary conditions is explicitly given by $p(x,y)=0$ and $\mathbf{u}(x,y)={((y^2-y)exp\left(y\right),(x^2-x)exp\left(x\right))}^T$. The obtained numerical results for the case where $\alpha =0$ are given in Table 1 and those for the case where $\alpha \ne 0$ are given in Table 2. The rate of convergence for the electric field $\mathbf{u}$ in the $\left|\left|\left|·\right|\right|\right|$ norm and in the $L^2$-norm is calculated in the fifth and seventh columns, respectively, and the rate of convergence for the Lagrange multiplier is written in the third column. It is clear that these results corroborate with the theoretical convergence estimates given in (22) and (23).

Table 1. Numerical table for example 1 ($\alpha =0$).

h	$\parallel {\mathit{\epsilon }}_{\mathit{h}}\parallel$	Rate	$\left|\left|\left|{\mathbf{e}}_{\mathit{h}}\right|\right|\right|$	Rate	$\parallel {\mathbf{u}}_{\mathit{h}}^0-{\mathbb{Q}}_{\mathit{h}}\left({\mathbf{u}}^0\right)\parallel$	Rate
2.5000 × ${10}^{-1}$	9.7222 × ${10}^{-3}$	-	6.8626 × ${10}^{-1}$	-	7.2247 × ${10}^{-2}$	-
1.2500 × ${10}^{-1}$	2.3299 × ${10}^{-3}$	2.0610 × ${10}^0$	3.6323 × ${10}^{-1}$	9.1786 × ${10}^{-1}$	1.8181 × ${10}^{-2}$	1.9905 × ${10}^0$
6.2500 × ${10}^{-2}$	5.7839 × ${10}^{-4}$	2.0101 × ${10}^0$	1.8435 × ${10}^{-1}$	9.7843 × ${10}^{-1}$	4.5607 × ${10}^{-3}$	1.9951 × ${10}^0$
3.1250 × ${10}^{-2}$	1.4436 × ${10}^{-4}$	2.0024 × ${10}^0$	9.2524 × ${10}^{-2}$	9.9456 × ${10}^{-1}$	1.1413 × ${10}^{-3}$	1.9986 × ${10}^0$
1.5625 × ${10}^{-2}$	3.6074 × ${10}^{-5}$	2.0006 × ${10}^0$	4.6306 × ${10}^{-2}$	9.9864 × ${10}^{-1}$	2.8539 × ${10}^{-4}$	1.9996 × ${10}^0$
7.8125 × ${10}^{-3}$	9.0176 × ${10}^{-6}$	2.0002 × ${10}^0$	2.3158 × ${10}^{-2}$	9.9966 × ${10}^{-1}$	7.1351 × ${10}^{-5}$	1.9999 × ${10}^0$

Table 2. Numerical table for example 1 ($\alpha =1$).

h	$\parallel {\mathit{\epsilon }}_{\mathit{h}}\parallel$	Rate	$\left|\left|\left|{\mathbf{e}}_{\mathit{h}}\right|\right|\right|$	Rate	$\parallel {\mathbf{u}}_{\mathit{h}}^0-{\mathbb{Q}}_{\mathit{h}}\left({\mathbf{u}}^0\right)\parallel$	Rate
2.5000 × ${10}^{-1}$	1.3732 × ${10}^{-2}$	-	6.8509 × ${10}^{-1}$	-	7.2632 × ${10}^{-2}$	-
1.2500 × ${10}^{-1}$	3.3933 × ${10}^{-3}$	2.6515 × ${10}^0$	3.6282 × ${10}^{-1}$	9.1702 × ${10}^{-1}$	1.8266 × ${10}^{-2}$	1.9915 × ${10}^0$
6.2500 × ${10}^{-2}$	8.4512 × ${10}^{-4}$	2.0054 × ${10}^0$	2.0168 × ${10}^0$	9.7833 × ${10}^{-1}$	4.5814 × ${10}^{-3}$	1.9953 × ${10}^0$
3.1250 × ${10}^{-2}$	2.1105 × ${10}^{-4}$	2.0016 × ${10}^0$	9.2428 × ${10}^{-2}$	9.9454 × ${10}^{-1}$	1.1464 × ${10}^{-3}$	1.9987 × ${10}^0$
1.5625 × ${10}^{-2}$	5.2746 × ${10}^{-5}$	2.0004 × ${10}^0$	4.6258 × ${10}^{-2}$	9.9863 × ${10}^{-1}$	2.8667 × ${10}^{-4}$	1.9997 × ${10}^0$
7.8125 × ${10}^{-3}$	1.3185 × ${10}^{-5}$	2.0001 × ${10}^0$	2.3134 × ${10}^{-2}$	9.9966 × ${10}^{-1}$	7.1671 × ${10}^{-5}$	1.9999 × ${10}^0$

4.2. Example 2

As in this test case, we choose a current density J so that the true solution of (4)–(6) is accorded by $\mathbf{u}(x,y)={(sin(y^2-y)e^y,sin(x^2-x)e^x)}^T$ and $p(x,y)$ is zero everywhere on ${(0,1)}^2$. The results for the cases where $\alpha =0$ and $\alpha =1$ are displayed in Table 3 and Table 4. Also, in this example the numerical results consolidate the errors estimations (22) and (23).

Table 3. Numerical table for example 2 ($\alpha =0$).

h	$\parallel {\mathit{\epsilon }}_{\mathit{h}}\parallel$	Rate	$\left|\left|\left|{\mathbf{e}}_{\mathit{h}}\right|\right|\right|$	Rate	$\parallel {\mathbf{u}}_{\mathit{h}}^0-{\mathbb{Q}}_{\mathit{h}}\left({\mathbf{u}}^0\right)\parallel$	Rate
2.5000 × ${10}^{-1}$	9.7235 × ${10}^{-3}$	-	6.7923 × ${10}^{-1}$	-	7.1752 × ${10}^{-2}$	-
1.2500 × ${10}^{-1}$	2.3313 × ${10}^{-3}$	2.0603 × ${10}^0$	3.5962 × ${10}^{-1}$	9.1742 × ${10}^{-1}$	1.8055 × ${10}^{-2}$	1.9906 × ${10}^0$
6.2500 × ${10}^{-2}$	5.7882 × ${10}^{-4}$	2.0100 × ${10}^0$	1.8254 × ${10}^{-1}$	9.7829 × ${10}^{-1}$	4.5293 × ${10}^{-3}$	1.9951 × ${10}^0$
3.1250 × ${10}^{-2}$	1.4447 × ${10}^{-4}$	2.0023 × ${10}^0$	9.1616 × ${10}^{-2}$	9.9452 × ${10}^{-1}$	1.1334 × ${10}^{-3}$	1.9986 × ${10}^0$
1.5625 × ${10}^{-2}$	3.6103 × ${10}^{-5}$	2.0006 × ${10}^0$	4.5852 × ${10}^{-2}$	9.9863 × ${10}^{-1}$	2.8342 × ${10}^{-4}$	1.9996 × ${10}^0$
7.8125 × ${10}^{-3}$	9.0249 × ${10}^{-6}$	2.0001 × ${10}^0$	2.2931 × ${10}^{-2}$	9.9966 × ${10}^{-1}$	7.0860 × ${10}^{-5}$	1.9999 × ${10}^0$

Table 4. Numerical table for example 2 ($\alpha =1$).

h	$\parallel {\mathit{\epsilon }}_{\mathit{h}}\parallel$	Rate	$\left|\left|\left|{\mathbf{e}}_{\mathit{h}}\right|\right|\right|$	Rate	$\parallel {\mathbf{u}}_{\mathit{h}}^0-{\mathbb{Q}}_{\mathit{h}}\left({\mathbf{u}}^0\right)\parallel$	Rate
2.5000 × ${10}^{-1}$	1.3723 × ${10}^{-2}$	-	6.7808 × ${10}^{-1}$	-	7.2139 × ${10}^{-2}$	-
1.2500 × ${10}^{-1}$	3.3929 × ${10}^{-3}$	2.0160 × ${10}^0$	3.5922 × ${10}^{-1}$	9.1659 × ${10}^{-1}$	1.8141 × ${10}^{-2}$	1.9915 × ${10}^0$
6.2500 × ${10}^{-2}$	8.4513 × ${10}^{-4}$	2.0053 × ${10}^0$	1.8235 × ${10}^{-1}$	9.7819 × ${10}^{-1}$	4.5500 × ${10}^{-3}$	1.9953 × ${10}^0$
3.1250 × ${10}^{-2}$	2.1106 × ${10}^{-4}$	2.0015 × ${10}^0$	9.1521 × ${10}^{-2}$	9.9450 × ${10}^{-1}$	1.1386 × ${10}^{-3}$	1.9987 × ${10}^0$
1.5625 × ${10}^{-2}$	5.2749 × ${10}^{-5}$	2.0004 × ${10}^0$	4.5804 × ${10}^{-2}$	9.9862 × ${10}^{-1}$	2.8471 × ${10}^{-4}$	1.9997 × ${10}^0$
7.8125 × ${10}^{-3}$	1.3186 × ${10}^{-5}$	2.0001 × ${10}^0$	2.2908 × ${10}^{-2}$	9.9966 × ${10}^{-1}$	7.1181 × ${10}^{-5}$	1.9999 × ${10}^0$

5. Conclusions and Remarks

In this paper, we presented and analyzed a new numerical scheme of the weak Galerkin mixed finite element technique for finding an approximation of the electrostatic field which is a solution of modified Maxwell’s equations by introducing a Lagrange multiplier. We proved that it is well posed, and established and tested numerically some optimal rates of convergence. Our results in this paper are satisfactory and encourage their application to other systems of PDEs, such as, for example, reaction–diffusion equations and Stokes equations.