Example 10/18 - 2D Maxwell equations using Nedelec elements

XLiFE++ provides Nedelec elements (first and second family) that are H(curl) comforming. Consider the following academic Maxwell problem:
{ curlcurlE − ω2μ𝜀E = f in Ω E × n = 0  on ∂Ω

with the following weak form:

| find E ∈ V = {v ∈ H(curl,Ω),v × n = 0 on ∂Ω} such that  ∫ ΩcurlEcurlv = ∫ ΩEv∀ ⁡v ∈ V.

Using first family Nedelec’s element, the XLiFE++ program looks like:

 
#include "xlife++.h" 
using namespace xlifepp; 
 
Real omg=1, eps=1, mu=1, a=pi_, ome=omg* omg* mu* eps; 
 
Vector<Real> f(const Point& P, Parameters& pa = defaultParameters) 
{ 
  Real x=P(1), y=P(2); 
  Vector<Real> res(2); 
  Real c=2*a*a-ome; 
  res(1)=-c*cos(a*x)*sin(a*y); 
  res(2)= c*sin(a*x)*cos(a*y); 
  return res; 
} 
 
Vector<Real> solex(const Point& P, Parameters& pa = defaultParameters) 
{ 
  Real x=P(1), y=P(2); 
  Vector<Real> res(2); 
  res(1)=-cos(a*x)*sin(a*y); 
  res(2)= sin(a*x)*cos(a*y); 
  return res; 
} 
 
int main(int argc, char** argv) 
{ 
  init(argc, argv, _lang=en); 
// mesh square using gmsh 
  SquareGeo sq(_xmin=0, _xmax=1, _ymin=0, _ymax=1, _nnodes=50, _side_names="Gamma"); 
  Mesh mesh2d(sq, triangle, 1, gmsh); 
  Domain omega=mesh2d.domain("Omega"); 
  Domain gamma=mesh2d.domain("Gamma"); 
// define space and unknown 
  Space V(_domain=omega, _FE_type=Nedelec, _order=1, _name="V"); 
  Unknown e(V, "E"); 
  TestFunction q(e, "q"); 
// define forms, matrices and vectors 
  BilinearForm aev=intg(omega, curl(e)|curl(q)) - ome*intg(omega, e|q); 
  LinearForm l=intg(omega, f|q); 
  EssentialConditions ecs = (ncross(e)|gamma=0); 
// compute 
  TermMatrix A(aev, ecs, "A"); 
  TermVector b(l, "B"); 
// solve 
  TermVector E=directSolve(A, b); 
// P1 interpolation, L2 projection on H1 
  Space W(_domain=omega, _interpolation=P1, _name="W"); 
  TermVector EP1=projection(E, W, 2); 
  EP1.name("E"); 
  saveToFile("E", EP1, vtu); 
  return 0; 
}

As Nedelec finite elements approximation are not conforming in H1, the solution is not continuous across elements (only tangent component is continuous). So to represent the solution, it is projected on H1 as follows:

| find E1 ∈ L2(Ω) such that ∫ ΩE1w = ∫ ΩEw∀ ⁡w ∈ L2(Ω).

Using an H1 conforming approximation for E1 leads to a continuous representation of the projection. We show on the next figure the Ex component field provided by this example:

pict pict

Figure 6.1: First component of the solution of the Maxwell 2D problem using Nedelec first family elements, and nodal error

pict

Figure 6.2: L2 errors versus the step h for 1 and 2 order Nedelec first family approximation