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Static Diffusion Problem Theory

Governing equation​

−∇⋅ν(∇u)=f-\nabla\cdot\nu\left(\nabla u\right)=f

where,

  • ν:=ν(x)\nu:=\nu(\mathbf{x}) is the diffusivity coefficient
  • f=f(x)f=f(\mathbf{x}) is the source term.

Boundary condition​

Dirichlet boundary condition​

u=g(x), on Γgu = g(x), \text{ on } \Gamma_{g}

Neumann boundary condition​

ν∇u⋅n=h, on Γh\nu \nabla u\cdot\boldsymbol{n}=h, \text{ on } \Gamma_{h}

where, h=h(x)h=h(\mathbf{x}) is the incoming flux.

Mixed boundary condition​

au+ν∇u⋅n=c, on Γmixau+\nu\nabla u\cdot\boldsymbol{n}=c, \text{ on } \Gamma_{mix}

Weak Dirichlet boundary condition​

  • TODO add description of weak boundary condition.

Galerkin FEM​

Variation form​

∫Ω∇w⋅ν∇udΩ−∫ΓmixawudS=∫ΓhwhdS+∫ΓmixwcdS+∫ΩwfdΩ\int_{\Omega}\nabla w\cdot\nu\nabla u {d\Omega}-\int_{\Gamma_{mix}}{awu}dS=\int_{\Gamma_{h}}whdS+\int_{\Gamma_{mix}}wcdS+\int_{\Omega}wfd\Omega

where, ww is vanishes at the Dirichlet boundary.

Discretized form​