Completely integrally closed Prufer v-multiplication domains

We study the effects on D of assuming that the power series ring D[[X]] is a v-domain or a Prufer v-multiplication domain (PVMD). We show that a PVMD D is completely integrally closed if and only if boolean AND(infinity)(n=1)(I-n)(v) = (0) for every proper t-invertible t-ideal I of D. Using this, we show that if D is an almost GCD domain, then D[[X]] is integrally closed if and only if D is a completely integrally closed PVMD with torsion t-class group. We also determine several classes of PVMDs for which being Archimedean is equivalent to being completely integrally closed and give some new characterizations of integral domains related to Krull domains.