2D Laplace problem with Dirichlet condition#

Keywords: PDE:Laplace Method:FEM FE:Lagrange Condition:Dirichlet Solver:Direct

Let us consider now the case of non-homogeneous Dirichlet condition on the boundaries \(x=0\) (\(\Sigma^-\)) and \(x=1\) (\(\Sigma^+\)):

\[u=1 \mathrm{\;on\;}\Sigma^-\cup \Sigma^+.\]

The variational formulation is now (\(a=0\))

\[\begin{split}\left| \begin{array}{c} \mathrm{find\;} u\in V=\left\{v\in L^2(\Omega), \nabla v\in (L^2(\Omega))^2\right\} \mathrm{\;such\;that} \\ \begin{aligned} & \displaystyle \int_\Omega \nabla u.\nabla v =\int_\Omega f\, v & & \forall v\in V,\ v=0 \mathrm{\;on\;}\Sigma^-\cup \Sigma^+\\ & u=1 & & \mathrm{\;on\;}\Sigma^-\cup \Sigma^+ \end{aligned} \end{array} \right.\end{split}\]

Its approximation by P1 Lagrange finite element is implemented in XLiFE++ as follows:

#include "xlife++.h"
using namespace xlifepp;

Real f(const Point& P, Parameters& pa = defaultParameters)
{return -8.;}

int main(int argc, char** argv)
{
  init(argc, argv, _lang=en); // mandatory initialization of xlifepp

  // create mesh of square
  Strings sn("y=0", "x=1", "y=1", "x=0");
  SquareGeo sq(_origin=Point(0., 0.), _length=1, _nnodes=20, _domain_name="Omega", _side_names=sn);
  Mesh mesh2d(sq, _shape=triangle, _generator=structured);
  Domain omega=mesh2d.domain("Omega");
  Domain sigmaM=mesh2d.domain("x=0"), sigmaP=mesh2d.domain("x=1");

  // create interpolation
  Space V(_domain=omega, _interpolation=P1, _name="V");
  Unknown u(V, _name="u");
  TestFunction v(u, _name="v");

  // create bilinear form, linear form and their algebraic representation
  BilinearForm auv=intg(omega, grad(u)|grad(v));
  LinearForm fv=intg(omega, f*v);
  EssentialConditions ecs= (u|sigmaM = 1) & (u|sigmaP = 1);
  TermMatrix A(auv, ecs, _name="A");
  TermVector B(fv, _name="B");

  // solve linear system AX=B
  TermVector U=directSolve(A, B);
  saveToFile("U_LD", U, _format=vtu);
  return 0;
}

../_images/ex_2d_dirichlet.png — Fig. 6 Solution of the Laplace 2D problem with Dirichlet condition on the unit square \([0,1]^2\)#

Note how easy is to take into account essential conditions. Only two lines has to be modified!