AMS691/529: FiniteElementMethods:...

AMS 691/529: Finite Element Methods:Theory and Computations

Lecture 14: More on Maxwell’s Equations

Xiangmin Jiao

SUNY Stony Brook

Xiangmin Jiao Finite Element Methods 1 / 12

Variational Formulation of Maxwell’s EquationsTo discretize time-harmonic Maxwell’s equation,

∇× (∇× E )− κ2εrE = iωµ0J , (1)

introduce test space V and its conjugate V∫Ω

(∇× (∇× E ) · v − κ2εrE · v

)dx =

∫Ωiωµ0J · v dx (2)

with essential BC E × n = 0 on Γ = ∂EΩ and natural BC(∇× E )× n = 0 on Σ = ∂Ω\∂EΩApply “integration by parts” and we obtain∫Ω∇×(∇×E )·v dx =

∫Ω

(∇×E )·(∇×v) dx+

∫∂Ω

((∇× E )× n)·v dS ,

where boundary integral vanishes if v = 0 or (∇× E )× n = 0Resulting variational form is∫

Ω

((∇× E ) · (∇× v)− κ2εrE · v

)dx =

∫Ωiωµ0J · v dx ,

Maxwell’s eq. is well-posed but noncoercive (i.e., indefinite) if κ > 0Xiangmin Jiao Finite Element Methods 2 / 12

Well-Posedness of Mixed Forms of Maxwell’s EquationsLet B = −i/ (ωµ0)∇× E (magnetic flux density), (1) then becomes

−iωµ0B + ∇× E = 0−∇× B + ωε0εrE = −J

Introducing test spaces Φ and Ψ, we obtain mixed variational forms

a(B,ψ) + b(E ,ψ) = 0

−b(B,φ) + c(E ,φ) =

∫Ω−J · φ dx

where

a(B,ψ) =

∫Ω−iωB ·ψ dx

b(E ,ψ) =

∫Ω∇× E ·ψ dx =

∫Ω∇×ψ · E dx

XXXXXXXXXX−∫∂Ω

(E × n) ·ψ dS

c(E ,φ) =

∫Ωωε0εrE · φ dx

This is not saddle-point problem if κ 6= 0, so LBB does not applyXiangmin Jiao Finite Element Methods 3 / 12

Well-Posedness of Mixed Forms of Maxwell’s EquationsLet E be in X ∈ H(curl; Ω) and u × n = 0 on ∂ΩE for u ∈ XLet B be in Y ∈ H(curl; Ω) and v × n = 0 on ∂Ω\∂ΩE for v ∈ YTheorem: Mixed formulation of Maxwell’s eq. is well-posed if κ 6= 0Basic idea of proof: Let A([E ; B] , [e; b]) denote bilinear form ofmixed form. For simplicity, assume µ0 = 1 and ωε0εr = 1. For any Eand B, choose test functions

e = (1 + i)E − (1− i)∇× B,

b = (1− i)B +1− i

ω∇× E ,

and then

A([E ; B] , [e; b]) = ‖E‖2L2 + ω‖B‖2L2 + ‖∇× B‖2L2 +1ω‖∇× E‖2L2

≥ min

ω,

1ω

‖[E ; B]‖L2 ≥ c ‖[E ; B]‖L2 ‖[e; b]‖L2

Hence, BNB1 holds; BNB2 can be shown similarlyCorollary: (1) is well posed with proper choice of test spaces in ∞-D


Difficulties in Finite Dimensions

If κ = 0, algebraic equation is singular due to semicoercivity in ∞-DI If E is solution, so is E + ∇f ∀f ∈ C 2 because ∇× (∇f ) = 0

If κ ≈ 0, algebraic equation can be extremely ill-conditionedIf κ 0, the presence of mass matrix∫

Ωκ2εrE · v dx

may prevent FEM discretization from producing singular systems, soBNB condition may be satisfiedHowever, standard Lagrange elements sometimes produce spurious(nonphysical) solutions even when κ 6= 0In particular, solution does not satisfy ∇ · (ωε0εrE ) = 0 when J = 0(i.e., source-free), violating identity ∇ · (∇× E ) = 0


Explanations of Spurious Solutions with Lagrange ElementsWhy well-posedness in ∞-D did not imply correct sol. in finite dim.?There are three explanations in the literature

I Missing divergence constraint1 −∇ · (εE ) = ∇ · J/(ωε0)F It makes sense physicallyF Why is this constraint not needed in ∞-D?F How to impose it correctly in a discrete sense?

I Curl has a large null space if ∇ · E is not constrained2

F This explanation makes sense for generalized eigenvalue problem∇× (∇× E) = λE when J = 0, due to spurious null spaces

F Why does it affect nonsingular algebraic equation when κ 0?I Lagrange elements are not C 1 continuous3

F This suggests why divergence constraint may get lostF In variational form∫

Ω

((∇× E) · (∇× v)− κ2εrE · v

)dx =

∫Ω

iωµ0J · v dx ,

where is the requirement of C 1 continuity for well-posed?1B. Jiang, J. Wu, and L.A. Povinelli. J. Comput. Phy. 125 (1996).2P. Monk, Finite Element Methods for Maxwell’s Equations, Oxford, 2003.3JM Jin, The Finite Element Method in Electromagnetics, 3rd ed., Wiley 2015. §5.8


Algebraic Explanation of Spurious SolutionsAn algebraic explanation of spurious solution with Lagrange elements:

I Test funcs. ψ = ψI =[[ψ, 0, 0]T , [0, ψ, 0]T , [0, 0, ψ]T

]may filter out

components in R (∇× (∇× ·)) (i.e., incompatability of spaces)I It is analogous to (but different from) violation of LBB condition,

which is only for well-posedness of saddle-point problemI Violation of this analogy of LBB condition does not cause violation of

BNB condition, but it has led to nonphysical solutionI If ψ = ∇ψ for ψ ∈ H1

0 , then spaces would have been compatible4

Fundamental reason in functional analysisI BNB condition ⇔ Hadamard well-posedness (existence and uniqueness)I Uniqueness 6⇒ correctness when bilinear forms have null spaces N

Fundamental question for PDEsI What is the necessary and sufficient condition for physically correct

solutions of PDEs in finite dimensions?I For advection-dominated PDEs, Hadamard well-posedness 6⇒ correct

solution in finite dimensions (later lectures)

4P. Monk, Finite Element Methods for Maxwell’s Equations, Oxford, 2003.Xiangmin Jiao Finite Element Methods 7 / 12

Algebraic Explanation of Spurious SolutionsAn algebraic explanation of spurious solution with Lagrange elements:

I Test funcs. ψ = ψI =[[ψ, 0, 0]T , [0, ψ, 0]T , [0, 0, ψ]T

]may filter out

components in R (∇× (∇× ·)) (i.e., incompatability of spaces)I It is analogous to (but different from) violation of LBB condition,

which is only for well-posedness of saddle-point problemI Violation of this analogy of LBB condition does not cause violation of

BNB condition, but it has led to nonphysical solutionI If ψ = ∇ψ for ψ ∈ H1

0 , then spaces would have been compatible4

Fundamental reason in functional analysisI BNB condition ⇔ Hadamard well-posedness (existence and uniqueness)I Uniqueness 6⇒ correctness when bilinear forms have null spaces N

Fundamental question for PDEsI What is the necessary and sufficient condition for physically correct

solutions of PDEs in finite dimensions?I For advection-dominated PDEs, Hadamard well-posedness 6⇒ correct

solution in finite dimensions (later lectures)4P. Monk, Finite Element Methods for Maxwell’s Equations, Oxford, 2003.


Possible Resolutions 1–3 of Spurious Solutions5

Resolution 1: for electrostatic field (with ∇× E = 0 andωε0εrE = −J), introduce φ s.t. E = −∇φ and solve

−∇ · (ε∇φ) =∇ · Jωε0

Resolution 2: use C 1 (i.e., H2) trial spacesI Would works if ψ = ∇ψ for ψ ∈ H1

0 (condition not given in Jin 2015)I Almost never used in practice due to its complexity

Resolution 3: utilize duality to filter out spurious solutionsI From mixed formulation, Maxwell’s equation can be converted to

uncoupled systems in B and EI If we are interested in B (or H), solve for E and then

B = −i/(ωµ0)∇× E , which filter out spurious modesI If we are interested in E , solve for B (or H) and then obtain EI Final solution has low-order accuracyI Does it always converge to correct solution?

5JM Jin, The Finite Element Method in Electromagnetics, 3rd ed., Wiley 2015. §5.8Xiangmin Jiao Finite Element Methods 8 / 12

Possible Resolution 4 of Spurious Solutions

Resolution 4: regularization with penalty termsI E.g., add ∫

Ω

s

[∇ · (εrE ) +

∇ · Jωε0

]2

dx (3)

as penalty term, where optimal s ≈ 1/µr ε2r

I As least-squares term, (3) improves/recovers coercivityI This reduces (but does not eliminate) spurious solutionsI May converge to incorrect solution when Ω has reentrant corners6I There are some more advanced alternatives7

6M. Costabel, M. Dauge, Nume. Math. 93 (2002) 239–2777A. Bonito et al., Math. Model. Numer. Anal., 50, 1457–1489, 2016.


Resolution 5: Edge-Based ElementsUse div-free spaces for trial spaces (and test space if Galerkin)

I All velocities are parallel to edgesI There is no flux on element boundary, so it is div-free within element

Idea was due to Whitney8 but its value was not recognized until 1980sBest-known divergence-free H(curl; Ω) spaces are due to Nédélec9

What if source term J 6= 0 so that ∇ · E 6= 0?

Answer: Helmholtz-decompose E to div-free and curl-free partsI Curl-free part is solution to Poisson equation as in Resolution 1I Dilemma: do we solve Poisson equations using edge elements?

8H. Whitney, Geometric Integration Theory, Princeton, 1957.9J.C. Nédélec, Numer. Meth., vol. 35, 1980.


Edge Elements and Finite Element Exterior CalculusEdge elements are the state of the art in FEM for Maxwell’s equationsGeneralization of Helmholtz decomposition to higher dimensions iscalled Hodge decomposition

I It generalizes Stokes theorem to closed Riemannian manifoldI It is built on top of De Rham cohomology in topology

Discrete analogy of De Rham cohomology on meshes is called finiteelement exterior calculus, with a discrete De Rham diagram


Discussions on Outstanding Challenges

Open problems in edge elements10

I arbitrary-order accuracy on hexahedral elements (due to technicalassumptions when deriving polynomials for tets)

I high-order spaces for pyramid elements (related to challenges forhexahedral elements)

Accuracy and efficiency of discretization schemesI Edge elements are less accurate and less efficient than Lagrange ones11

Significant challenge in engineering applications12

I Singularities/discontinuities across interfaces between different mediaI Edge elements do not enforce continuity normal to edges, but may still

lead to spurious oscillations (similar challenges at reentrant corners)Significant challenge in linear algebra

I Robust and efficient solvers for ill-conditioned/singular system as κ→ 0I One approach is to add stabilization (or penalty) terms

10L. Demkowicz, book chapter in Mixed Finite Elements, Springer, p. 152, 2008.11G. Mur, IEEE Trans. Magn. 30.5 (1994): 3552-3557.12JM Jin, The Finite Element Method in Electromagnetics, 3rd ed., Wiley 2015. §5.8


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