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Example 11/18 - Eigenvalues and eigenvectors of Laplace operator

This example shows how to get eigen functions of Laplace operator equipped with homogeneous Neumann condition:
{ −Δu + u = λu in Ω ∂ nu = 0  on ∂Ω

and its variational formulation in V = H1(Ω):

| find (u,λ) ∈ V × ℝ such that  ∫ Ω∇ ⁡u.∇ ⁡v + ∫ Ωuv = λ∫ Ωuv∀ ⁡v ∈ V.
 
#include "xlife++.h" 
using namespace xlifepp; 
 
int main(int argc, char** argv) 
{ 
  init(argc, argv, _lang=en); // mandatory initialization of xlifepp 
 
// mesh square 
  SquareGeo sq(_origin=Point(0., 0.), _length=1, _nnodes=20); 
  Mesh mesh2d(sq, triangle, 1, gmsh); 
  Domain omega = mesh2d.domain("Omega"); 
 
// build P2 interpolation 
  Space Vk(_domain=omega, _interpolation=P2, _name="Vk"); 
  Unknown u(Vk, "u"); 
  TestFunction v(u, "v"); 
 
// build eigen system 
  BilinearForm auv = intg(omega, grad(u) | grad(v)) + intg(omega, u * v) , 
               muv = intg(omega, u * v); 
  TermMatrix A(auv, "auv"), M(muv, "muv"); 
 
// compute the 10 first smallest in magnitude 
  EigenElements eigs = eigenInternSolve(A, M, _nev=10, _mode=krylovSchur, _which="SM");  // internal solver 
  theCout << eigs.values; 
  saveToFile("eigs", eigs.vectors, vtu); 
  return 0; 
}

pict

Figure 7.1: 9 first eigen vectors of the Laplace 2D problem with P2 elements