Abstract
A novel finite element method for the approximation of Maxwell’s equations over hybrid two-dimensional grids is studied. The choice of appropriate basis functions and numerical quadrature leads to diagonal mass matrices which allow for efficient time integration by explicit methods.On purely rectangular grids, the proposed schemes coincide with well-established FIT and FDTD methods. Additional internal degrees of freedom introduced on triangles allow for mass-lumping without the usual constraints on the shape of these elements. A full error analysis of the method is developed and numerical tests are presented for illustration.
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1 Introduction
The propagation of electromagnetic waves through a non-dispersive linear medium can be described by the time-dependent Maxwell’s equations
together with appropriate initial and boundary conditions. Here E, H denote the electric and magnetic field intensities, \(\varepsilon \), \(\mu \) the corresponding material parameters, and j describes the density of source and eddy currents. An efficient discretization of (1)–(2) can be achieved by the finite difference time domain (FDTD) method or the finite integration technique (FIT), see e.g. [14, 15], and for isotropic materials and orthogonal grids, second-order convergence can be obtained in space and time. In order to handle complex geometries, several attempts have been made to generalize these methods to non-orthogonal and unstructured grids; see e.g. [2, 12, 13] and also [3, 4] for more recent results. A rigorous error analysis of a Yee-like scheme on triangles and tetrahedra was given in [6], and first-order convergence in space on general unstructured grids was demonstrated theoretically and numerically.
Scope. In this paper, we propose a novel Yee-like discretization scheme for hybrid grids in two space dimensions, consisting of triangles and rectangles. The method is based on a finite element approximation with mass-lumping through numerical quadrature, which allows for a rigorous error analysis; see [5, 9] for background. On rectangular grid cells, the resulting discretization coincides with that of the FIT or FDTD method. Following [8], additional internal degrees of freedom are introduced on triangular grid cells, which allows us to prove discrete stability without severe restrictions on the mesh. The lowest order approximation on two-dimensional hybrid grids is studied in detail. The main ideas behind the construction of the method and its analysis however carry over to three dimensions and higher-order approximations; see [7, 8, 11] and the discussion at the end of the paper.
2 Description of the Problem
Let us start with completely specifying the model problem to be considered in the rest of the paper. We choose \(\varepsilon =\mu =1\) and abbreviate \(f=-\partial _{t}j\). Moreover, we consider the second-order form of Maxwell’s equations, i.e.,
with simple boundary conditions. The computational domain \(\varOmega \subseteq \mathbb {R}^2\) is assumed to be a bounded Lipschitz polygon and \({{\,\textrm{curl}\,}}E = \partial _x E_2 - \partial _y E_1 \) denotes the curl of a vector field \(E=(E_1,E_2)\) in two space dimensions. The above differential equations are considered on a finite time interval [0, T], and complemented by suitable initial conditions \(E(0)=E_0\) and \(\partial _{t}E(0) = F_0\). The existence of a unique solution can then be established by semi-group theory or Galerkin approximation. Solutions of (3)–(4) can further be characterized equivalently by the variational identities
for all \(v\in H({{\,\textrm{curl}\,}},\varOmega )= \{E\in L^2(\varOmega )^2\,:\, {{\,\textrm{curl}\,}}E\in L^2(\varOmega )\}\) and a.a. \(t \in [0,T]\). For abbreviation, we write \((a,b)=\int _\varOmega a \cdot b \, dx\) for the scalar product on \(L^2(\varOmega )\) and \(L^2(\varOmega )^2\).
3 A Finite Element Method with Mass-Lumping
Let \(\mathscr {T}_h\) = \(\{K\}\) be a geometrically-conforming quasi-uniform shape-regular partition of \(\varOmega \) into triangular and/or rectangular elements K. By assumption, all edges of the mesh are of similar length and we call the size h of the longest edge the mesh size.
Finite Element Spaces. For the approximation of the field E on individual elements, we consider local polynomial spaces defined by
Here \({{\,\mathrm{\mathscr {N}}\,}}_0(K)\) is the lowest order Nedelec space for triangles or rectangles [1, 10], and \({{\,\mathrm{\mathscr {B}}\,}}(K)\) is a space of three quadratic functions with vanishing tangential components. The corresponding degrees of freedom are depicted in Fig. 1, and details on the basis functions are presented in Sect. 5.The global finite element space is finally defined by \(\mathscr {V}_h= \{v_h\in H({{\,\textrm{curl}\,}};\varOmega )\,:\,\, v_h|_K\in V(K) \ \forall K \in \mathscr {T}_h\}.\)
Quadrature. We use an approximation \((u,v)_{h}:=\sum _{K}(u,v)_{h,K}\) for the \(L^2\)-scalar product, with contributions obtained by numerical integration. On the triangle, we set
where \(m_{K,i}\) is the midpoint of the edge \(e_i\) opposite to vertex i; see Fig. 1. For the rectangle, we proceed differently: Here we decompose \((u,v) = (u_1, v_1) + (u_2,v_2)\) into two contributions for the orthogonal directions, and then use different quadrature rules for the two contributions, i.e.
Here \(m_{K,h,i}\) and \(m_{K,h,j}\) are the midpoints of the horizontal and vertical edges, respectively; see again Fig. 1. For the semi-discretization of our model problem in space, we then consider the following inexact Galerkin approximation.
Problem 1
Let \(E_{h,0}\), \(E_{h,1} \in \mathscr {V}_h\) be given. Find \(E_h:[0,T]\rightarrow \mathscr {V}_h\) such that
for all \(v_h\in \mathscr {V}_h\) and all \(t\in [0,T]\), and such that \(E_h(0)=E_{h,0}\) and \(\partial _{t} E_h(0)=E_{h,1}\).
As we will indicate below, the implementation of this method leads to a diagonal mass matrix, which allows using explicit methods for efficient time integration.
4 Main Results
By elementary computations, one can verify the following assertions, which ensure the well-posedness of Problem 1 and yield a starting point for our error analysis.
Lemma 1
The quadrature rule (8) is exact for polynomials of degree \(k\le 2\) on triangles and for polynomials of degree \(k\le 1\) on squares. Moreover, the inexact scalar product \((\cdot ,\cdot )_h\) induces a norm \(\Vert \cdot \Vert _h\) on \(\mathscr {V}_h\), which is equivalent to the \(L^2\)-norm on \(\mathscr {V}_h\), and consequently Problem 1 has a unique solution.
As a second ingredient, let us recall some results about polynomial interpolation. We denote by \(\varPi _h : H^1(\mathscr {T}_h)^2 \rightarrow \mathscr {V}_h\) the projection defined element-wise by
where \(\varPi _K : H^1(K) \rightarrow {{\,\mathrm{\mathscr {N}}\,}}_0(K)\) is the standard interpolation operator for the lowest order Nedelec space \({{\,\mathrm{\mathscr {N}}\,}}(K)\) on both triangles and squares; see [1, 10] for details. We further denote by \(\pi _h^0 : L^2(\varOmega ) \rightarrow P_0(\mathscr {T}_h)\) the \(L^2\)-orthogonal projection onto piecewise constants; the same symbol is used for the projection of vector-valued functions.
Lemma 2
Let \(K\in \mathscr {T}_h\) and \(\varPi _h\) defined as in (10). Then
whenever E is regular enough, with a constant C independent of h.
Having introduced all the required tools, we can now state and prove our main result.
Theorem 1
Let E and \(E_h\) denote the solutions of (5) and (9) with initial values set by \(E_h(0)= \varPi _h E(0)\) and \(\partial _t E_h(0) = \varPi _h \partial _t E(0)\). Then
with constant C depending on the norm of E but independent of the mesh size h.
Proof
Apart from some technical details, the analysis follows by standard arguments. For completeness and convenience of the reader, we present all the details.
Step 1. Error Splitting and Estimate for the Projection Error. We first split the discretization error into a projection error and a discrete error component via
By the estimates of Lemma 2, we immediately obtain
which already covers the first error component.
Step 2. Discrete Error Equation. By subtracting (8) from (5) with \(v=v_h\), we can see that the discrete error \(\psi _h\) satisfies the identity
for all \(v_h \in \mathscr {V}_h\) and \(0 \le t \le T\), with quadrature error
We can further split \(\sigma _h(E,\phi ) = \sum \nolimits _{K\in \mathscr {T}_h}\sigma _K(E,\phi )\) into element contributions defined by \(\sigma _K(E,\phi ) = (E,\phi )_{h,K}-(E,\phi )_K\). Moreover, \(\psi _h(0)=\partial _{t} \psi _h(0)=0\), due to the choice of initial conditions for the discrete problem.
Step 3. Estimates for the Quadrature Error. To further proceed in our analysis, we now quantify the local quadrature error in more detail.
Lemma 3
Let \(E \in L^2(\varOmega )^2\) with \(E|_K \in H^1(K)^2\) for all \(K \in \mathscr {T}_h\). Then
for all \(\phi _h\in \mathscr {V}_h\) and all \(K\in \mathscr {T}_h\) with constant C independent of the element K.
Proof
Using Lemma 1, we deduce that \((u_h^0,v_h)_K=(u_h^0,v_h)_{h,K}\) for all \(u_h^0 \in P_0(K)^2\) and \(v_h \in V(K)\). We can then estimate the quadrature error by
where we used the Cauchy-Schwarz inequality and the norm equivalence of Lemma 1 and the approximation properties of the projections from Lemma 2.
Step 4. Estimate for the Discrete Error. Taking \(v_h=\partial _t \psi _h(t)\) as test function in the discrete error equation and integrating from 0 to t leads to
The three terms can now be estimated separately. Using Cauchy-Schwarz and Young inequalities, the first term may be bounded by
For the second term, we utilize that
The third term can finally be estimated using Lemma 3 according to
Using these estimates in the inequality (16), absorbing all the terms with the test function into the left side, and taking the supremum over \(t\in [0,T]\), after applying the norm equivalence of Lemma 1 to some terms, then leads to the estimate
for the discrete error component; one may also take the square root in all terms.
Step 5. The proof of the theorem is completed by applying the triangle inequality to the error splitting in Step 1 and adding up the estimates for the projection error \(\eta \) and the discrete error component \(\psi _h\).
5 Implementation
For completeness of the presentation, let us briefly discuss the choice of basis functions for the local finite element spaces \({{\,\mathrm{\mathscr {N}}\,}}_0(K)\) and \({{\,\mathrm{\mathscr {N}}\,}}_0^+(K)\) which, together with the numerical quadrature leads to diagonal mass matrices.
Rectangle. On quadrilateral elements K, we choose the standard basis for the lowest order Nedelec space \({{\,\mathrm{\mathscr {N}}\,}}_0(K)=\text {span}\{\varPhi _{h,i}, \varPhi _{v,i}: i=1,\ldots ,2\}\); see [1, 10]. These functions have the following properties: The function \(\varPhi _{h,i}\) associated to a horizontal edge \(e_{h,i}\) vanishes identically on the opposite horizontal edge, and \(\varPhi _{v,j}\) associated to for the vertical edge \(e_{v,j}\) vanishes on the opposite vertical edge. Hence the local mass matrix produced by the quadrature rule \((u,v)_{K,h}\) for every rectangle is diagonal.
Triangle. Let \(\{\lambda _i\}\) be the barycentric coordinates of the element K. For every edge \(e_k=e_{ij}\) pointing from vertex i to j, and thus opposite to k, we define the two basis functions
Then \({{\,\mathrm{\mathscr {N}}\,}}_0^+(K)={\text {span}}\{\varPhi _{12},\varPhi _{23},\varPhi _{31},\varPhi _{12}^{B},\varPhi _{23}^{B},\varPhi _{31}^{B}\}\). The bubble functions \(\varPhi _{ij}^{B}\) have vanishing tangential components on the edge are \(e_k\), and they vanish identically on the two remaining edges \(e_i\), \(e_j\). The functions \(\varPhi _{ij}\) are modified Nedelec basis functions. They have vanishing tangential components on the two edges \(e_i\), \(e_j\), and by appropriate choice of the parameters \(\alpha _{ij}\), \(\beta _{ij}\), \(\gamma _{ij}\), their normal components on all edge midpoints \(m_{K,i}\) can be made zero. As a consequence, the local mass matrix produced by the scalar product \((u,v)_{K,h}\) for the triangle becomes diagonal.
Summary. The global mass matrix is obtained by assembling the local mass matrices, which are diagonal, and hence has inherits this property.
6 Numerical Illustration
We consider the computational domain \(\varOmega = \varOmega _1\cup \varOmega _2\) where \(\varOmega _1 = (0,2)\times (-1,1)\) and \(\varOmega _2 = \big ((2,4)\times (-1,1))\setminus B_{0.3}(3,0)\), where \(B_{r}(x,y)\) denotes the ball with radius r around midpoint (x, y). The two subdomains are meshed by rectangles and triangles, respectively. For our test problem, we consider the wave Eq. (3). The boundary \(\partial \varOmega \) is split into several parts and as boundary conditions, we impose
The initial conditions are chosen as \(E(0)=\partial _{t}E(0)=0\). This corresponds to a pulse entering at the left boundary, propagating through the domain, and getting reflected at the walls of the box and the circular inclusion. Some snapshots of the solution are depicted in Fig. 2. Let us remark that no reflections are observed at the interface between the two meshes. In our numerical tests, we observe linear convergence of the error in space. This coincides with the theoretical predictions of Theorem 1, and also demonstrates that the error estimates are sharp. Note that second-order convergence is in general lost for Yee-like approximations on unstructured grids; also see [11].
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Egger, H., Radu, B. (2024). A Yee-Like Finite Element Scheme for Maxwell’s Equations on Hybrid Grids with Mass-Lumping. In: van Beurden, M., Budko, N.V., Ciuprina, G., Schilders, W., Bansal, H., Barbulescu, R. (eds) Scientific Computing in Electrical Engineering. SCEE 2022. Mathematics in Industry(), vol 43. Springer, Cham. https://doi.org/10.1007/978-3-031-54517-7_9
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