Dirichlet Problem

In the Dirichlet problem the voltage on the boundary is prescribed. Given $f\in H^{1/2}(\partial \Omega )$, find $u\in H^1(\Omega )$ with

$$\nabla \cdot (\gamma \nabla u)=0\text{in }\Omega ,u=f\text{on }\partial \Omega .$$

Weak form. Find $u\in H^1(\Omega )$ with $\mathrm{tr}u=f$ such that

$${\int }_{\Omega }\gamma \nabla u\cdot \nabla v\mathrm{d}x=0\forall v\in H_0^1(\Omega ).$$

Writing $u=u_0+Ef$ with any extension $Ef\in H^1(\Omega )$ of $f$ turns this into a problem for $u_0\in H_0^1(\Omega )$. That problem has a unique solution by the Lax-Milgram Theorem; coercivity comes from the Poincaré inequality on $H_0^1$.

The measured quantity is the resulting boundary current $g=\gamma {\partial }_{\nu }u{|}_{\partial \Omega }\in H^{-1/2}(\partial \Omega )$. The map $f\mapsto g$ is the Dirichlet-to-Neumann Map.

By the Dirichlet principle, $u$ minimises the power ${\int }_{\Omega }\gamma |\nabla v{|}^2$ among all $v$ with trace $f$.

In a finite element code, the boundary condition is imposed on the matrix as described in Enforcing Dirichlet Conditions.

In ModularEIT.jl: forward_dirichlet.

References

1. L. C. Evans (2010). Partial Differential Equations, 2nd ed. AMS GSM 19. doi:10.1090/gsm/019
2. L. Borcea (2002). Electrical impedance tomography. Inverse Problems 18(6), R99–R136. doi:10.1088/0266-5611/18/6/201