We solve the correlation diffusion equation by the finite element method.^{21}^{–}^{23} Equation (7) is multiplied by a test function which obeys the boundary conditions, and whose zeroth and first derivatives are integrable over the domain. The boundary conditions of Eq. (9) are incorporated by the subsequent integration by parts. The domain is subdivided into a mesh of nonoverlapping elements joined at $N$ vertex nodes. On this mesh, we define a set of piecewise linear basis functions such that $ui(rj)=\delta ij$ for $i,j=1,\u2026,N$ where $rj$ located at the $j$’th vertex node. We subsequently approximate the solution $G(r,\tau )\u2248\u2211jNuj(r)Gj(\tau )$. Selecting the basis functions in the weak formulation to be the same as the mesh basis allows us to write the resulting linear system of equations: Display Formula
$A(\tau )[x]G(\tau )=q,$(12)
where $A$ is the finite element system matrix. We express the parameters of the forward model, $x(r)$, the absorption-like decorrelation function $h(r)$, and the diffusion coefficient $\kappa $ using the same basis functions such that, for example, $\mu a(r)=\u2211ku\mu ,k(r)\mu a,k$. Consequently, Display Formula$A(\tau )[x]ij=\u222b\Omega \u2211k[u\kappa ,k(r)\kappa k\u2207ui(r)\xb7\u2207uj(r)+[u\mu ,k(r)\mu a,k+uh,k(r)hk(\tau )]ui(r)uj(r)]dnr+12A\u222b\delta \Omega ui(r)uj(r)dn\u22121r.$(13)
and Display Formula$qj=\u222b\Omega qjuj(r)dnr.$(14)