Institut de Recherche MAthématique de Rennes

LMS Durham Symposium: Computational methods for wave propagation in direct scattering. - July, Durham, UK The hp version of the Weighted Regularization Method for Maxwell Equations Martin COSTABEL & Monique DAUGE Institut de Recherche MAthématique de Rennes http://www.maths.univ-rennes.fr/~costabel

References Theory and analysis of the h -version: M. Costabel, M. Dauge: Weighted regularization of Maxwell equations in polyhedral domains. Numer. Math. Online publication DOI.7/s88 () Numerical tests, including hp version, benchmark problems: http://www.maths.univ-rennes.fr/~dauge/benchmax.html Numerical results in D: M. Costabel, M. Dauge, D. Martin, G. Vial: Weighted regularization of Maxwell equations computations in curvilinear polygons. Proceedings of ENUMATH () Our Finite Element package: D. Martin: Mélina. Online documentation: http://www.maths.univ-rennes.fr/~dmartin Analysis of the hp version: M. Costabel, M. Dauge, Ch. Schwab: Exponential convergence of the hp -FEM for the weighted regularization of Maxwell equations in polygonal domains. In preparation

Convergence of Eigenvalues:

The Ideal... Valeur propre Neumann n sur le Maillage - Interp Interp 7 8 9 - - -8 - Interp Interp Interp Interp Interp 7 Interp 8 7 8 9 7 8 9 7 8 9 9 9 st Neumann eigenvalue on a curved rectangle Nodal elements Two elements p version Interp 9 - Relative error vs # DOF

... and the reality Valeur propre Maxwell n pénalisée au bord par λ = ; Maillage Interp Interp Interp st Maxwell Interp Interp 7 8 9 eigenvalue on a curved L shape Nodal elements Regularization without weight - Relative error vs # DOF

The standard nodal approximation of the first singularity The exact solution ( grad(r sin θ )). Computation with Q elements, regularization without weight. Non- H singularities are not approximated.

The Ideal Recovered - - Geo. deg. Geo. deg. 7 8 9 st Maxwell eigenvalue on a curved L shape Geo. deg. 7 Nodal elements - Geo. deg. Geo. deg. Geo. deg. 8 8 9 9 9 Weighted regularization -8 Relative error vs # DOF

Outline The Maxwell source problem and eigenproblem and their variational formulation with weighted regularization. Convergence analysis of the weighted regularization method: The h version The hp version Numerical results in D: Straight L-shaped domain Curved L-shaped domain Evidence for exponential convergence Numerical results in D: The thick L Fichera s corner. 7

Maxwell source problem Permittivity ε and permeability µ set to. Ω polyhedron in R (or polygon in R ). Given ω R, J L (Ω) with div J =, find (E, H) L (Ω) such that curl E iωh = & curl H + iωe = J in Ω, (Maxwell SP) E n = & H n = on Ω Maxwell eigenvalue problem Find non-zero ω such that (E, H) curl E iωh = & curl H + iωe = (Maxwell EVP) E n = & H n = on Ω in Ω, In both cases, div E = div B = is implied if ω. 8

Variational formulation:the WRM Bilinear form for s >, < α, a s (u, v) = Ω curl u curl v + s Ω rα div u div v, where r(x) dist(x, { nonconvex edges and corners }). Find u X such that v X : a s (u, v) ω Ω u v = Ω f v (SP): ω, f given, u unknown; (EVP): f = ; u, ω unknown. Energy space X = X γ N := H (curl) {u L (Ω), r γ div u L (Ω)}. with γ = α/. Recall: σ(p [s] rγ ) = σ(m) s σ(ldir r ). γ 9

Galerkin approximation: Everything standard here With the test and trial subspace X h X, the Galerkin solution u h X h is defined by v h X h : a s (u h, v h ) ω Ω u h v h = Ω f v h Since a s (, ) is positive definite and symmetric, we have the usual stability and error estimates, in particular: Céa s lemma: For the source problem and ω = : u u h X γ N inf{ u v h X γ N v h X h } Convergence of the eigenvalues: For the eigenvalue problem, f = : Suppose {X h } h> is a family of subspaces approximating X. If ω σ(p [s] r γ discrete problem such that ω ω h = O((energy norm error) )). The rest is approximation theory... is a simple eigenvalue, then there is a simple eigenvalue ω h of the

Birman-Solomyak decompositions Basic idea for the estimate of the approximation error:. Decomposition of the solution u = w + grad ϕ, with w regular and ϕ carrying the main corner singularities Energy norm: u X γ N w H + grad ϕ L + rγ ϕ L. Approximation of u = w + grad ϕ by u h = w h + grad ϕ h, u u h X γ N w w h H + ϕ ϕ h V γ, where w h X h, and ϕ h belongs to an auxiliary space Φ h satisfying Note: grad Φ h X h. Since X h consists of globally continuous functions, Φ h must consist of globally C functions. The space Φ h is not used in the computations, but only in the estimation of the approximation error.

B-S decomposition suitable for the h version Suitable hypotheses on the weight, smooth right hand side (or EV problem), D,... Theorem: The solution can be decomposed as u = w + grad ϕ, with w H +τ (Ω) and ϕ m= K m β Here (D, one reentrant corner of opening ω ) H +τ is the regularity of the Neumann problem for the Laplace operator in Ω, i.e. τ < π/ω, and β is determined by the regularity of the Dirichlet problem, β > π/ω. The weighted Sobolev spaces K m β ϕ K m β are defined by the seminorms = α =m r β+ α α ϕ L Result: Energy norm estimate of order O(h τ ) for any mesh that allows C elements.

B-S decomposition suitable for the hp version At present: Ω has to be a polygon in R. Weighted spaces of analytic functions, compare BABUŠKA-GUO: A β (Ω) m= K m β (Ω), C : m : ϕ K m β Cm+ m! One assumes that the right hand side is analytic (true for EVP), for example in A (Ω). Theorem: For γ (γ Ω, ] there is a δ γ > such that for all < δ < δ γ, the solution u can be decomposed as u = w + grad ϕ, with w A δ (Ω) and ϕ H A γ δ(ω) Result: For suitably (geometrically) refined meshes, one has an energy norm error estimate of order O(e bp ) for some b >.

For the analysis of the hp version, types of exponentially refined meshes: In CYAN, the patch to consider for the construction of C interpolants.

Rectangles with hanging nodes.. -. - -. -. - -...

Q -quadrilaterals.. -. - -. -. - -...

Triangles.. -. - -. -. - -... 7

Problems with Q -quadrilaterals Polynomials on the reference square K are not polynomials on the physical element K, in general. The Jacobian J P ( K) is not constant, therefore the gradient (in physical space) does not map Q p ( K) into Q p ( K). The (auxiliary) approximation FE space Φ h for the potentials has to be chosen not as Q p ( K), but as J m Q p m ( K) (depends on K!), for some m. There is a geometrical condition on the patches (the jacobians J on both sides have to be proportional on the interface) which is not always satisfied, but satisfied in our case of the L and two axiparallel sides. a a b b a b = a b 8

Computations in D The straight L: Ω is the L -shaped domain : [., ] [., ] \ [.7, ] [.7, ]. Computation of the first eigenvalues Λ[s] for s = [ : ], α =,,, q =,,, The computed eigenvalues are sorted depending on whether curl u s r α/ div u is ρ or ρ with a fixed ρ. 9

Legend For the next 8 figures. In Cyan, the Laplace-Neumann eigenvalues, which coincide for D domains with Maxwell eigenvalues. In Blue, the circles represent the computed Λ[s] with curl -dominant eigenvectors. In Red, the lines join the computed Λ[s] with div -dominant eigenvectors. Abscissa : s / Ordinate : Λ[s].

q =, α = 7 8 9

q =, α = 7 8 9 7

q =, α = 7 8 9 8

Geometrical mesh refinement, on straight L-shape domain (sl) Ratio... -. -. - - -. - -... -. - -... layers. layers. 9

Geometrical mesh refinement, on straight L-shape domain (sl) Ratio... -. -. - - -. - -... -. - -... layers. layers.

First Maxwell ev. on L-domain, with α = and ref. ratio On domain (sl) with number of layers m & funct. interp. p. - - Nb of layers Nb of layers Nb of layers Nb of layers 7 8 7 9 8 7 9 8 7 9 8 7 9 8 7 9 8 7 9 8 7 9 8 9 Relative error versus the number of DOF. Computed with MÉLINA FEM library Nb of layers 7 developped by - Nb of layers 8 D. MARTIN Nb of layers 9 (Rennes, ENSTA). -8 Nb of layers

First Maxwell ev. on L-domain, with α = and ref. ratio On domain (sl) with number of layers m & funct. interp. p. - - - -8 7 8 9 Nb of layers 7 8 9 Nb of layers 7 8 9 Nb of layers 7 8 9 Nb of layers 7 8 9 7 Nb of layers 7 77 8 9 Nb of layers 8 8 8 9 Nb of layers 9 9 Nb of layers Relative error versus the number of DOF. Computed with MÉLINA FEM library developped by D. MARTIN (Rennes, ENSTA).

Second Maxwell ev. on L-domain, with α = and ref. ratio On domain (sl) with number of layers m & funct. interp. p. - - - -8 - Nb of layers Nb of layers Nb of layers Nb of layers Nb of layers 7 Nb of layers 8 Nb of layers 9 Nb of layers 7 8 9 7 8 9 7 8 9 7 8 9 7 8 9 7 8 9 7 8 9 7 8 9 Relative error versus the number of DOF. Computed with MÉLINA FEM library developped by D. MARTIN (Rennes, ENSTA).

Second Maxwell ev. on L-domain, with α = and ref. ratio On domain (sl) with number of layers m & funct. interp. p. - - - -8 - Nb of layers Nb of layers Nb of layers Nb of layers Nb of layers 7 Nb of layers 8 Nb of layers 9 Nb of layers 7 8 9 7 8 9 7 8 9 7 7 7 77 8 8 8 9 89 8 9 9 9 Relative error versus the number of DOF. Computed with MÉLINA FEM library developped by D. MARTIN (Rennes, ENSTA).

Curved L-shaped domain..... The curved L-shaped domain (cl) with a π/ corner.

Meshes on the approximate curved L-domain.......... Ω q,m, q =, m =,

First Maxwell ev. on curved L, with weight & refinement On domain (cl) with ratio, number of layers m =, and α =. - - - Geo. deg. Geo. deg. Geo. deg. Geo. deg. Geo. deg. Geo. deg. 7 8 9 9 7 8 8 Geom. interp. n & Funct. interp. p. Relative error versus the number of DOF. Computed with MÉLINA. -8 8 7

Second Maxwell ev. on curved L, with weight & refinement On domain (cl) with ratio, number of layers m = and α =. - - - -8 - - 7 8 9 Geo. deg. 7 9 8 Geo. deg. Geo. deg. Geo. deg. Geo. deg. Geo. deg. Geom. interp. n & Funct. interp. p. Relative error versus the number of DOF. Computed with MÉLINA. Something weird going on here... 8

First Maxwell ev. on curved L, with weight & refinement On domain (cl) with ratio, number of layers m =, and α =. - - Geo. deg. Geo. deg. 7 8 9 Nodes = Gauss-Lobatto points. Geom. interp. n & Funct. interp. Geo. deg. 7 p. - Geo. deg. Geo. deg. 8 8 9 Relative error Geo. deg. 9 9 versus the # of DOF. -8 9

Second Maxwell ev. on curved L, with weight & refinement On domain (cl) with ratio, number of layers m = and α =. - - - -8 - - Geo. deg. Geo. deg. Geo. deg. 7 8 9 7 8 9 Geo. deg. 7 Geo. deg. Geo. deg. 7 8 9 8 9 Nodes = Gauss-Lobatto points. Geom. interp. n & Funct. interp. p. Relative error versus the # of DOF....this is much better!

Exponential convergence Domain (sl) with ratio, α =, lowest eigenvalues. - - 7 eigenvalue No eigenvalue No eigenvalue No eigenvalue No h p version: - 8 9 eigenvalue No p = #layers -8-7 7 7 8 8 8 9 9 9 Relative error vs #DOF - 7 8 -

Exponential convergence Domain (sl) with ratio, α =, lowest eigenvalues...8. 7 8 7 8 9 7 7 9 8 7 h p version: p = #layers.. eigenvalue No eigenvalue No eigenvalue No log(-log(error)) vs eigenvalue No eigenvalue No log(#dof) -. -....8....8.

Exponential convergence Domain (sl) with ratio, α =, lowest eigenvalues. - - 7 eigenvalue No eigenvalue No eigenvalue No eigenvalue No eigenvalue No h p version: - 8 9 p = #layers -8-7 7 7 8 8 8 9 9 9 log(error) vs (#DOF) / - 7 8-8 8

D hp version: Computations are working nicely, Theory: not yet finished... In the following graphs, we show results for two D domains: The thick L L [, ], where L = [, ] \ [, ] is the D straight L-shaped domain. Fichera s corner (7/8th cube) [, ] \ [, ]

The Thick L: Legend The Maxwell eigenvalues of the thick L L [, ] are known: They are sums of Dirichlet and Neumann eigenvalues of the D domain L (which can easily be computed to high precision) and of the interval [, ] (which are of the form π n, n N ). The following 9 graphs show the error as a function of the number of degrees of freedom for the first 8 Maxwell eigenvalues and computed convergence rates for the th eigenvalue. The computations were done for Q p elements on a Q -hexahedral mesh, obtained by using a geometrically refined mesh on L with refinement ratio 8 and a variable number of layers indicated by color, and two layers in the third dimension. The degree p varies from to. For the weight sr α, we used s =, α =, and r the distance to the singular edge {(, )} [, ].

The Thick L: Maxwell Eigenvalue vp Maxwell (D) nº -- Ratio de maillage 8 - - Nb of layers Nb of layers Nb of layers 7 7 rel. error vs #dof - - Nb of layers Nb of layers Nb of layers Nb of layers 7 p #layers 7 - -

The Thick L: Maxwell Eigenvalue vp Maxwell (D) nº -- Ratio de maillage 8 - - Nb of layers Nb of layers Nb of layers Nb of layers Nb of layers Nb of layers Nb of layers 7 7 7 rel. error vs #dof p #layers 7 - - 7

The Thick L: Maxwell Eigenvalue vp Maxwell (D) nº -- Ratio de maillage 8-7 - Nb of layers Nb of layers rel. error vs #dof - - Nb of layers Nb of layers Nb of layers Nb of layers Nb of layers 7 7 p #layers 7 - - 8

The Thick L: Maxwell Eigenvalue vp Maxwell (D) nº -- Ratio de maillage 8 - - - - - convergence rates p #layers 7 - -7-8

The Thick L: Maxwell Eigenvalue vp Maxwell (D) nº -- Ratio de maillage 8 - Nb of layers Nb of layers rel. error vs #dof - - Nb of layers Nb of layers Nb of layers Nb of layers Nb of layers 7 7 7 p #layers 7 - -

The Thick L: Maxwell Eigenvalue vp Maxwell (D) nº -- Ratio de maillage 8 - - - Nb of layers Nb of layers Nb of layers Nb of layers Nb of layers Nb of layers Nb of layers 7 7 7 rel. error vs #dof p #layers 7-8 -

The Thick L: Maxwell Eigenvalue 7 vp Maxwell (D) nº 7 -- Ratio de maillage 8 - - - Nb of layers Nb of layers Nb of layers Nb of layers Nb of layers Nb of layers Nb of layers 7 7 7 rel. error vs #dof p #layers 7-8 -

The Thick L: Maxwell Eigenvalue 8 vp Maxwell (D) nº 8 -- Ratio de maillage 8 - - Nb of layers Nb of layers Nb of layers 7 7 rel. error vs #dof - - Nb of layers Nb of layers Nb of layers Nb of layers 7 p #layers 7 - -

Fichera Corner The eigenvalues for this domain are not known. Therefore we show the validity of our computations with the weighted regularization method in two ways: Dependence of the eigenvalues on the parameter s in the weight sr α : We compute the first Maxwell eigenvalues for s ranging from to. The curl dominant eigenvalues are shown in blue, the div dominant eigenvalues in red, and the indifferent ones in green (factor ρ =. ). We see that for α = the eigenvalues show the required behavior, i.e. they lie on two families of straight lines. Without weight ( α = ), the behavior is different. Approximation of the cube. We deform the 7/8th cube so that it approximates the unit cube [, ]. The Maxwell eigenvalues for the latter are known ( π times sums of squares of integers). In our graphs, we show the eigenvalues divided by π to make this approximation more evident.

Fichera Corner cont d We show computations for domains: F, the 7/8th cube [, ] \ [, ]. The eigenvalues have high multiplicity due to symmetries. F = F ([.8, ] [., ] [, ]) Here the symmetries are broken. From F to F and from F to F, the thickness of the protruding hexahedra is reduced by a factor of, so that F is % away from the unit cube [, ] in Hausdorff distance. We use Q hexahedral elements on a tensor product mesh generated from the -dimensional mesh {, /, /, } [, ]. For readability reasons, in the following picture the refinement ratio / is replaced by / and the geometric approximation factor / by /.

Fichera Corner F [a,b,c] = [,,] Fichera Corner F [a,b,c] = [.,.8,] The domains F... F - - - - - - Fichera Corner F [a,b,c] = [.,.,.] Fichera Corner F [a,b,c] = [.,.,.] - - - - - - 7

Ratio de tri. et α =. Domaine F F, no weight: α =...8... 8

Ratio de tri. et α =. Domaine F F, α =...8... 9

Ratio de tri. et α =. Domaine F F, α =...

7 Ratio de tri. et α =. Domaine F F, α =

8 Ratio de tri. et α =. Domaine F F, α = 7

Conclusion The Weighted Regularizaton Method requires only a very simple modification of the standard regularized variational formulation of time-harmonic Maxwell source and eigenvalue problems. It allows the use of standard nodal finite elements even in the case of domains with reentrant corners and solutions with strong non- H singularities where the standard regularization gives incorrect results. In combination with the hp version of the finite element method, an efficient and robust algorithm is obtained that allows highly precise approximations even for D problems.