Discontinuous Galerkin Methods with Time-Operators in Their Numerical Traces for Time-Dependent Electromagnetics

Bernardo Cockburn; Shukai Du; Manuel A. Sánchez

doi:10.1515/cmam-2021-0215

Article

Discontinuous Galerkin Methods with Time-Operators in Their Numerical Traces for Time-Dependent Electromagnetics

Bernardo Cockburn , Shukai Du and Manuel A. Sánchez

Published/Copyright: May 20, 2022

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From the journal Computational Methods in Applied Mathematics Volume 22 Issue 4

Abstract

We present a new class of discontinuous Galerkin methods for the space discretization of the time-dependent Maxwell equations whose main feature is the use of time derivatives and/or time integrals in the stabilization part of their numerical traces. These numerical traces are chosen in such a way that the resulting semidiscrete schemes exactly conserve a discrete version of the energy. We introduce four model ways of achieving this and show that, when using the mid-point rule to march in time, the fully discrete schemes also conserve the discrete energy. Moreover, we propose a new three-step technique to devise fully discrete schemes of arbitrary order of accuracy which conserve the energy in time. The first step consists in transforming the semidiscrete scheme into a Hamiltonian dynamical system. The second step consists in applying a symplectic time-marching method to this dynamical system in order to guarantee that the resulting fully discrete method conserves the discrete energy in time. The third and last step consists in reversing the above-mentioned transformation to rewrite the fully discrete scheme in terms of the original variables.

Keywords: Time-Dependent Maxwell’s Equations; Discontinuous Galerkin Methods; Energy-Conserving Methods

MSC 2010: 78M10; 65M22

Dedicated to the memory of Francisco-Javier Sayas

Funding source: National Science Foundation

Award Identifier / Grant number: 1912646

Funding source: Fondo Nacional de Desarrollo Científico y Tecnológico

Award Identifier / Grant number: 11180284

Funding statement: B. Cockburn was supported in part by the NSF through grant DMS-1912646. M. A. Sánchez was supported by FONDECYT Iniciación grant n. 11180284.

A Fully Discrete Methods: Second Choice of DG Traces

A.1 ESPRK Methods

We have that

$( ϵ ⁢ E h n , i , v ) T h + ( F h n , i , ∇ × v ) T h - ⟨ n × Φ ^ ⁢ ( - F h n , i , E h n , i ) , v ⟩ ∂ ⁡ T h = - ( J n , i , v ) T h , ( μ ⁢ H h n , i - H n , i - 1 b i ⁢ Δ ⁢ t , r ) T h + ( E h n , i , ∇ × r ) T h - ⟨ n × Ψ ^ ⁢ ( E h n , i , - F h n , i ) , r ⟩ ∂ ⁡ T h = 0 , ( F h n , i - F h n , i - 1 b ~ i - 1 ⁢ Δ ⁢ t , v ) T h + ( H h n , i - 1 , v ) T h = 0$

for all $( v , w ) ∈ V h × W h$ .

As a consequence, each step of the new fully discrete DG scheme reads

$( ϵ ⁢ E h n , i - E h n , i - 1 b ~ i - 1 ⁢ Δ ⁢ t , v ) T h - ( H h n , i - 1 , ∇ × v ) T h - ⟨ n × Φ ^ ⁢ ( H h n , i - 1 , E h n , i - E h n , i - 1 b ~ i - 1 ⁢ Δ ⁢ t ) , v ⟩ ∂ ⁡ T h = - ( J n , i - J n , i - 1 b ~ i - 1 ⁢ Δ ⁢ t , v ) T h , ( μ ⁢ H h n , i - H n , i - 1 b i ⁢ Δ ⁢ t , r ) T h + ( E h n , i , ∇ × r ) T h - ⟨ n × Ψ ^ ⁢ ( E h n , i , - F h n , i ) , r ⟩ ∂ ⁡ T h = 0$

for all $( v , w ) ∈ V h × W h$ , and where $F h n , i = F h n - Δ ⁢ t ⁢ ∑ j = 1 i - 1 b ~ j ⁢ H h n , j$ .

We still have to provide the definition of $( E h 0 , F h 0 )$ . Since typically the initial value given is $E 0$ , we must define $( E h 0 , F h 0 ) ∈ V h × W h$ at once in terms of the initial data $F 0$ . This is done in a way similar to the previous case. Let us note that such definition must be such that $F h 0 ∈ W h$ satisfies the equation

$( ϵ ⁢ E h 0 , v ) T h + ( F h 0 , ∇ × v ) T h - ⟨ n × Φ ^ ⁢ ( - F h 0 , E h 0 ) , v ⟩ ∂ ⁡ T h = 0 for all ⁢ v ∈ V h .$

A.2 Symplectic DIRK Methods

By similar arguments used in Section 4.5, we obtain the symplectic DIRK time-marching scheme for the 𝑯-𝑭 formulation as follows:

$1 a i ⁢ i ⁢ Δ ⁢ t ⁢ ( μ ⁢ H h n , i , r ) T h + ( E h n , i , ∇ × r ) T h + ⟨ n × Ψ ^ ⁢ ( E h n , i , - F h n , i ) , r ⟩ ∂ ⁡ T h = ( μ ⁢ R H n , i , r ) T h , ( ϵ ⁢ E h n , i , v ) T h + ( F h n , i , ∇ × v ) T h - ⟨ n × Φ ^ ⁢ ( - F h n , i , E h n , i ) , v ⟩ ∂ ⁡ T h = - ( J n , i , v ) T h , 1 a i ⁢ i ⁢ Δ ⁢ t ⁢ ( F h n , i , r ) T h + ( H h n , i , r ) T h = ( R F n , i , r ) T h ,$

where $R G n , i$ is defined by (A.1).

We next rewrite the scheme in terms of the 𝑬-𝑯 formulation. First, we multiply the second equation by $( a i ⁢ i ⁢ Δ ⁢ t ) - 1$ and obtain

$1 a i ⁢ i ⁢ Δ ⁢ t ⁢ ( ϵ ⁢ E h n , i , v ) T h + ( R F n , i - H h n , i , ∇ × v ) T h + ⟨ n × Φ ^ ⁢ ( R F n , i - H h n , i , - 1 a i ⁢ i ⁢ Δ ⁢ t ⁢ E h n , i ) , v ⟩ ∂ ⁡ T h = ( - J n , i a i ⁢ i ⁢ Δ ⁢ t , v ) T h .$

Observing that

$Φ ^ ⁢ ( R F n , i - H h n , i , - 1 a i ⁢ i ⁢ Δ ⁢ t ⁢ E h n , i ) = Φ ^ ⁢ ( R F n , i , 0 ) - Φ ^ ⁢ ( H h n , i , E h n , i a i ⁢ i ⁢ Δ ⁢ t ) ,$

we obtain

$1 a i ⁢ i ⁢ Δ ⁢ t ⁢ ( ϵ ⁢ E h n , i , v ) T h - ( H h n , i , ∇ × v ) T h - ⟨ n × Φ ^ ⁢ ( H h n , i , 1 a i ⁢ i ⁢ Δ ⁢ t ⁢ E h n , i ) , v ⟩ ∂ ⁡ T h = Θ ,$

where

$Θ = - ( R F n , i , ∇ × v ) T h - ⟨ n × Φ ^ ⁢ ( R F n , i , 0 ) , v ⟩ ∂ ⁡ T h - ( J n , i a i ⁢ i ⁢ Δ ⁢ t , v ) T h .$

Note that

$( ϵ ⁢ R E n , i , v ) T h + ( R F n , i , ∇ × v ) T h - ⟨ n × Φ ^ ⁢ ( - R F n , i , R E n , i ) , v ⟩ ∂ ⁡ T h = - ( R J n , i , v ) T h ,$

which implies that

$Θ = ( ϵ ⁢ R E n , i , v ) T h - ⟨ n × Φ ^ ⁢ ( 0 , R E n , i ) , v ⟩ ∂ ⁡ T h - ( J ⁢ ( t n , i ) a i ⁢ i ⁢ Δ ⁢ t , v ) T h + ( R J n , i , v ) T h .$

Thus, the DIRK scheme for the 𝑬-𝑯 formulation reads as follows.

Perform the following computations for $i = 1 , … , s$ :
1. Compute the right-hand-side terms for $G = E h , H h , F h , J$ ,
  (A.1) $R G n , i = G n a i ⁢ i ⁢ Δ ⁢ t + ∑ j = 1 i - 1 a i ⁢ j a i ⁢ i ⁢ ( G n , j a j ⁢ j ⁢ Δ ⁢ t - R G n , j ) ,$
  where $J n , i = J ⁢ ( t n , i )$ and $t n , i = t n + c i ⁢ Δ ⁢ t$ .
2. Compute the solution $( E h n , i , H h n , i )$ of the system
  $1 a i ⁢ i ⁢ Δ ⁢ t ⁢ ( μ ⁢ H h n , i , r ) T h + ( E h n , i , ∇ × r ) T h + ⟨ n × Ψ ^ ⁢ ( E h n , i , - F h n , i ) , r ⟩ ∂ ⁡ T h = ( μ ⁢ R H n , i , r ) T h , 1 a i ⁢ i ⁢ Δ ⁢ t ⁢ ( ϵ ⁢ E h n , i , v ) T h - ( H h n , i , ∇ × v ) T h - ⟨ n × Φ ^ ⁢ ( H h n , i , 1 a i ⁢ i ⁢ Δ ⁢ t ⁢ E h n , i ) , v ⟩ ∂ ⁡ T h = Θ n , i ,$
  with $F h n , i = a i ⁢ i ⁢ Δ ⁢ t ⁢ ( R F n , i - H h n , i )$ and
  $Θ n , i := ( ϵ ⁢ R E n , i , v ) T h - ⟨ n × Φ ^ ⁢ ( 0 , R E n , i ) , v ⟩ ∂ ⁡ T h - ( J ⁢ ( t n , i ) a i ⁢ i ⁢ Δ ⁢ t , v ) T h + ( R J n , i , v ) T h .$
Advance the solution $G n + 1 = G n + Δ ⁢ t ⁢ ∑ i = 1 s b i ⁢ k G i$ for $G = E h , F h , H h$ , where
$k G i = G n , i - G n a i ⁢ i ⁢ Δ ⁢ t - ∑ j = 1 i - 1 a i ⁢ j a i ⁢ i ⁢ k G j .$

B Fully Discrete Methods: Third Choice of DG Traces

B.1 ESPRK Methods

Based on the ESPRK scheme (4.1) and the semidiscrete Hamiltonian system (5.2), we obtain the fully discrete ESPRK scheme for the DG methods with the third choice of the numerical traces,

$( ϵ E h n , i - E h n , i - 1 b i ⁢ Δ ⁢ t , v ) T h + ⟨ C 11 ⟦ E h n , i - E h n , i - 1 b i ⁢ Δ ⁢ t ⟧ , ⟦ v ⟧ ⟩ F h = ( H h n , i , ∇ × v ) T h + ⟨ n × ( { { H h n , i } } + C ¯ 12 T ⟦ H h n , i ⟧ ) , v ⟩ ∂ ⁡ T h ∖ Γ + ⟨ n × H h n , i , v ⟩ Γ - ( J ( t n + c i Δ t ) , v ) T h ,$

$( μ H h n , i - H h n , i - 1 b ~ i - 1 ⁢ Δ ⁢ t , r ) T h + ⟨ C 22 ⟦ H h n , i - H h n , i - 1 b ~ i - 1 ⁢ Δ ⁢ t ⟧ , ⟦ r ⟧ ⟩ F h 0 = - ( E h n , i - 1 , ∇ × r ) T h - ⟨ { { E h n , i - 1 } } + C ¯ 12 ⟦ E h n , i - 1 ⟧ , r ⟩ ∂ ⁡ T h ∖ Γ - ⟨ g E ( t n + c ~ j - 1 Δ t ) , r ⟩ Γ .$

B.2 Symplectic DIRK Methods

Here, we apply the symplectic DIRK scheme summarized in Section 4.5 to the semidiscrete system (5.2). Then the full-discrete scheme can be summarized in the following steps.

Perform the following computations for $i = 1 , … , s$ :
1. Compute the right-hand-side terms for $G = E h , H h$ ,
  $R G n , i = G n a i ⁢ i ⁢ Δ ⁢ t + ∑ j = 1 i - 1 a i ⁢ j a i ⁢ i ⁢ ( G n , j a j ⁢ j ⁢ Δ ⁢ t - R G n , j ) .$
2. Recall that $J n , i = J ⁢ ( t n , i )$ , $g E n , i = g E ⁢ ( t n , i )$ and $t n , i = t n + c i ⁢ Δ ⁢ t$ . Compute the solution $( E h n , i , H h n , i )$ ,
  $1 a i ⁢ i ⁢ Δ ⁢ t ( ( ϵ E h n , i , v ) T h + ⟨ C 11 ⟦ E h n , i ⟧ , ⟦ v ⟧ ⟩ F h ) = ( H h n , i , ∇ × v ) T h + ⟨ n × ( { { H h n , i } } + C ¯ 12 T ⟦ H h n , i ⟧ ) , v ⟩ ∂ ⁡ T h ∖ Γ + ⟨ n × H h n , i , v ⟩ Γ - ( J n , i , v ) T h + ( ϵ R E n , i , v ) T h + ⟨ C 11 ⟦ R E n , i ⟧ , ⟦ v ⟧ ⟩ F h ,$
  
  $1 a i ⁢ i ⁢ Δ ⁢ t ( ( μ H h n , i , r ) T h + ⟨ C 22 ⟦ H h n , i ⟧ , ⟦ r ⟧ ⟩ F h 0 ) = - ( E h n , i , ∇ × r ) T h - ⟨ { { E h n , i } } + C ¯ 12 ⟦ E h n , i ⟧ , r ⟩ ∂ ⁡ T h ∖ Γ - ⟨ g E n , i , r ⟩ Γ + ( μ R H n , i , r ) T h + ⟨ C 22 ⟦ R H n , i ⟧ , ⟦ r ⟧ ⟩ F h .$
Advance the solution $G n + 1 = G n + Δ ⁢ t ⁢ ∑ i = 1 s b i ⁢ k G i$ for $G = E h , H h$ , where
$k G i = G n , i - G n a i ⁢ i ⁢ Δ ⁢ t - ∑ j = 1 i - 1 a i ⁢ j a i ⁢ i ⁢ k G j .$

C Fully Discrete Methods: Fourth Choice of DG Traces

C.1 ESPRK Methods

Based on the ESPRK scheme (4.1) and the semidiscrete Hamiltonian system (6.1), we obtain the fully discrete ESPRK scheme for the DG methods with the fourth choice of the numerical traces,

$( ϵ ⁢ E h n , i - E h n , i - 1 b i ⁢ Δ ⁢ t , v ) T h = ( H h n , i , ∇ × v ) T h + ⟨ n × ( { { H h n , i } } + C ¯ 12 T ⟦ H h n , i ⟧ ) , v ⟩ ∂ ⁡ T h ∖ Γ + ⟨ n × H h n , i , v ⟩ Γ - ( J ( t n + c i Δ t ) , v ) T h - ⟨ C 11 ⟦ A h n , i ⟧ , ⟦ v ⟧ ⟩ F h ,$

$⟨ C 22 ⟦ F h n , i - F h n , i - 1 b i ⁢ Δ ⁢ t ⟧ , ⟦ r ⟧ ⟩ F h 0 = ⟨ C 22 ⟦ H h n , i ⟧ , ⟦ r ⟧ ⟩ F h 0 ,$

$( μ ⁢ H h n , i - H h n , i - 1 b ~ i - 1 ⁢ Δ ⁢ t , r ) T h = - ( E h n , i - 1 , ∇ × r ) T h - ⟨ n × ( { { E h n , i - 1 } } + C ¯ 12 ⟦ E h n , i - 1 ⟧ ) , r ⟩ ∂ ⁡ T h ∖ Γ - ⟨ g E ( t n + c ~ j - 1 Δ t ) , r ⟩ Γ - ⟨ C 22 ⟦ F h n , i - 1 ⟧ , ⟦ r ⟧ ⟩ F h 0 ,$

$⟨ C 11 ⟦ A h n , i - A h n , i - 1 b ~ i - 1 ⁢ Δ ⁢ t ⟧ , ⟦ v ⟧ ⟩ F h = ⟨ C 11 ⟦ E h n , i - 1 ⟧ , ⟦ v ⟧ ⟩ F h ,$

or, simply,

$F h n , i - F h n , i - 1 b i ⁢ Δ ⁢ t = H h n , i on ⁢ ∂ ⁡ T h ∖ Γ ,$

$( μ ⁢ H h n , i - H h n , i - 1 b ~ i - 1 ⁢ Δ ⁢ t , r ) T h = - ( E h n , i - 1 , ∇ × r ) T h - ⟨ n × ( { { E h n , i - 1 } } + C ¯ 12 ⟦ E h n , i - 1 ⟧ ) , r ⟩ ∂ ⁡ T h ∖ Γ - ⟨ g E ( t n + c ~ j - 1 Δ t ) , r ⟩ Γ - ⟨ C 22 ⟦ F h n , i - 1 ⟧ , ⟦ r ⟧ ⟩ F h ,$

$A h n , i - A h n , i - 1 b ~ i - 1 ⁢ Δ ⁢ t = E h n , i - 1 on ⁢ ∂ ⁡ T h ,$

$( A h n , i ) ext = g A ⁢ ( t n , i ) × n on ⁢ Γ .$

C.2 Symplectic DIRK Methods

Here, we apply the symplectic DIRK scheme summarized in Section 4.5 to the semidiscrete system (6.1). Then the full-discrete scheme can be summarized in the following steps.

Perform the following computations for $i = 1 , … , s$ :
1. Compute the right-hand-side terms for $G = E h , F h , H h , A h$ ,
  $R G n , i = G n a i ⁢ i ⁢ Δ ⁢ t + ∑ j = 1 i - 1 a i ⁢ j a i ⁢ i ⁢ ( G n , j a j ⁢ j ⁢ Δ ⁢ t - R G n , j ) .$
2. Recall that $J n , i = J ⁢ ( t n , i )$ , $g E n , i = g E ⁢ ( t n , i )$ and $t n , i = t n + c i ⁢ Δ ⁢ t$ . Compute the solution $( E h n , i , F h n , i , H h n , i , A h n , i )$ ,
  $1 a i ⁢ i ⁢ Δ ⁢ t ⁢ ( ϵ ⁢ E h n , i , v ) T h = ( H h n , i , ∇ × v ) T h + ⟨ n × ( { { H h n , i } } + C ¯ 12 T ⟦ H h n , i ⟧ ) , v ⟩ ∂ ⁡ T h ∖ Γ + ⟨ n × H h n , i , v ⟩ Γ - ( J n , i , v ) T h - ⟨ C 11 ⟦ A h n , i ⟧ , ⟦ v ⟧ ⟩ F h + ( ϵ R E n , i , v ) T h ,$
  
  $1 a i ⁢ i ⁢ Δ ⁢ t ⁢ F h n , i = H h n , i + R F n , i on ⁢ ∂ ⁡ T h ∖ Γ ,$
  
  $1 a i ⁢ i ⁢ Δ ⁢ t ⁢ ( μ ⁢ H h n , i , r ) T h = - ( E h n , i , ∇ × r ) T h - ⟨ n × ( { { E h n , i } } + C ¯ 12 ⟦ E h n , i ⟧ ) , r ⟩ ∂ ⁡ T h ∖ Γ - ⟨ g E n , i , r ⟩ Γ - ⟨ C 22 ⟦ F h n , i ⟧ , ⟦ r ⟧ ⟩ F h 0 + ( μ R H n , i , r ) T h ,$
  
  $1 a i ⁢ i ⁢ Δ ⁢ t ⁢ A h n , i = E h n , i + R A n , i on ⁢ ∂ ⁡ T h ,$
  
  $( A h n , i ) ext = g A ⁢ ( t n , i ) × n on ⁢ Γ .$
Advance the solution $G n + 1 = G n + Δ ⁢ t ⁢ ∑ i = 1 s b i ⁢ k G i$ for $G = E h , F h , H h , A h$ , where
$k G i = G n , i - G n a i ⁢ i ⁢ Δ ⁢ t - ∑ j = 1 i - 1 a i ⁢ j a i ⁢ i ⁢ k G j .$

References

[1] M. Ainsworth, Dispersive and dissipative behaviour of high order discontinuousGalerkin finite element methods, J. Comput. Phys. 198 (2004), no. 1, 106–130. 10.1016/j.jcp.2004.01.004Search in Google Scholar

[2] T. S. Brown, S. Du, H. Eruslu and F.-J. Sayas, Analysis of models for viscoelastic wave propagation, Appl. Math. Nonlinear Sci. 3 (2018), no. 1, 55–96. 10.21042/AMNS.2018.1.00006Search in Google Scholar

[3] B. Cockburn, The pursuit of a dream, Francisco Javier Sayas and the HDG methods, SeMA J. 79 (2022), no. 1, 37–56. 10.1007/s40324-021-00273-ySearch in Google Scholar

[4] B. Cockburn, Z. Fu, A. Hungria, L. Ji, M. A. Sánchez and F.-J. Sayas, Stormer–Numerov HDG methods for acoustic waves, J. Sci. Comput. 75 (2018), no. 2, 597–624. 10.1007/s10915-017-0547-zSearch in Google Scholar

[5] B. Cockburn, J. Gopalakrishnan and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal. 47 (2009), no. 2, 1319–1365. 10.1137/070706616Search in Google Scholar

[6] B. Cockburn, N. C. Nguyen and J. Peraire, HDG methods for hyperbolic problems, Handbook of Numerical Methods for Hyperbolic Problems, Handb. Numer. Anal. 17, Elsevier/North-Holland, Amsterdam (2016), 173–197. 10.1016/bs.hna.2016.07.001Search in Google Scholar

[7] B. Cockburn and V. Quenneville-Bélair, Uniform-in-time superconvergence of the HDG methods for the acoustic wave equation, Math. Comp. 83 (2014), no. 285, 65–85. 10.1090/S0025-5718-2013-02743-3Search in Google Scholar

[8] S. Du and F.-J. Sayas, A unified error analysis of hybridizable discontinuous Galerkin methods for the static Maxwell equations, SIAM J. Numer. Anal. 58 (2020), no. 2, 1367–1391. 10.1137/19M1290966Search in Google Scholar

[9] G. Fu and C.-W. Shu, Optimal energy-conserving discontinuous Galerkin methods for linear symmetric hyperbolic systems, J. Comput. Phys. 394 (2019), 329–363. 10.1016/j.jcp.2019.05.050Search in Google Scholar

[10] Z. Fu, L. F. Gatica and F.-J. Sayas, Algorithm 949: MATLAB tools for HDG in three dimensions, ACM Trans. Math. Software 41 (2015), no. 3, Article ID 20. 10.1145/2658992Search in Google Scholar

[11] J. Gopalakrishnan, M. Solano and F. Vargas, Dispersion analysis of HDG methods, J. Sci. Comput. 77 (2018), no. 3, 1703–1735. 10.1007/s10915-018-0781-zSearch in Google Scholar

[12] A. Hungria, D. Prada and F. J. Sayas, HDG methods for elastodynamics, Comput. Math. Appl. 74 (2017), 2671–2690. 10.1016/j.camwa.2017.08.016Search in Google Scholar

[13] N. C. Nguyen, J. Peraire and B. Cockburn, High-order implicit hybridizable discontinuous Galerkin methods for acoustics and elastodynamics, J. Comput. Phys. 230 (2011), no. 10, 3695–3718. 10.1016/j.jcp.2011.01.035Search in Google Scholar

[14] N. C. Nguyen, J. Peraire and B. Cockburn, Hybridizable discontinuous Galerkin methods for the time-harmonic Maxwell’s equations, J. Comput. Phys. 230 (2011), no. 19, 7151–7175. 10.1016/j.jcp.2011.05.018Search in Google Scholar

[15] M. A. Sánchez, C. Ciuca, N. C. Nguyen, J. Peraire and B. Cockburn, Symplectic Hamiltonian HDG methods for wave propagation phenomena, J. Comput. Phys. 350 (2017), 951–973. 10.1016/j.jcp.2017.09.010Search in Google Scholar

[16] M. A. Sánchez, B. Cockburn, N.-C. Nguyen and J. Peraire, Symplectic Hamiltonian finite element methods for linear elastodynamics, Comput. Methods Appl. Mech. Engrg. 381 (2021), Paper No. 113843. 10.1016/j.cma.2021.113843Search in Google Scholar

[17] J. M. Sanz-Serna, Symplectic Runge–Kutta and related methods: Recent results, Phys. D 60 (1992), 293–302. 10.1016/0167-2789(92)90245-ISearch in Google Scholar

[18] F.-J. Sayas, T. S. Brown and M. E. Hassell, Variational Techniques for Elliptic Partial Differential Equations: Theoretical Tools and Advanced Applications, CRC Press, Boca Raton, 2019. 10.1201/9780429507069Search in Google Scholar

[19] M. Stanglmeier, N. C. Nguyen, J. Peraire and B. Cockburn, An explicit hybridizable discontinuous Galerkin method for the acoustic wave equation, Comput. Methods Appl. Mech. Engrg. 300 (2016), 748–769. 10.1016/j.cma.2015.12.003Search in Google Scholar

Received: 2021-11-13

Revised: 2022-02-02

Accepted: 2022-03-09

Published Online: 2022-05-20

Published in Print: 2022-10-01

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DOI: doi.org/10.1515/cmam-2021-0215

Keywords for this article

Time-Dependent Maxwell’s Equations; Discontinuous Galerkin Methods; Energy-Conserving Methods