This model considers an SPDE over a domain Ω which is partitioned into k subdomains Ωd, d∈{1,…,k}, where ∪kd=1Ωd=Ω. A common marginal variance is assumed but the range can be particular to each Ωd, rd.
From Bakka et al. (2019), the precision matrix is Q=1σ2R˜C−1R for Rr=C+18k∑d=1r2dGd,˜Cr=π2k∑d=1r2d˜Cd where σ2 is the marginal variance. The Finite Element Method - FEM matrices: C, defined as Ci,j=⟨ψi,ψj⟩=∫Ωψi(s)ψj(s)∂s, computed over the whole domain, while Gd and ˜Cd are defined as a pair of matrices for each subdomain (Gd)i,j=⟨1Ωd∇ψi,∇ψj⟩=∫Ωd∇ψi(s)∇ψj(s)∂s and (˜Cd)i,i=⟨1Ωdψi,1⟩=∫Ωdψi(s)∂s.
In the case when r=r1=r2=…=rk we have Rr=C+r28G and ˜Cr=πr22˜C giving Q=2πσ2(1r2C˜C−1C+18C˜C−1G+18G˜C−1C+r264G˜C−1G) which coincides with the stationary case in Lindgren and Rue (2015), when using ˜C in place of C.
In practice we define rd as rd=pdr, for known p1,…,pk constants. This gives ˜Cr=πr22k∑d=1p2d˜Cd=πr22˜Cp1,…,pk and 18k∑d=1r2dGd=r28k∑d=1p2d˜Gd=r28˜Gp1,…,pk where ˜Cp1,…,pk and ˜Gp1,…,pk are pre-computed.