Shoichi Fujima

Robustness of a Characteristic Finite Element Scheme of Second Order in Time Increment

In using the characteristic finite element method we have to pay much attention to the errors caused by numerical integration. Compared to the first order scheme, the second order scheme is robust with respect to the numerical integration error, which has been recognized by numerical results. We have also given a rough explanation for the robustness. Under a mild condition on the numerical integration we can show that the second order scheme is stable with the numerical integration. A detail discussion and a complete proof of the stability will be presented in the forthcoming paper.

Note on the Class Number of the pth Cyclotomic Field, II

Abstract For an odd prime number p, let h−p denote the relative class number of the pth cyclotomic field . It is conjectured that h−p is odd when p is of the form p = 2ℓ + 1 with an odd prime number ℓ, and it is known that the conjecture is valid if 2 is a primitive root modulo ℓ. In this article, we handle a prime number p of the form p = 2e + 1ℓ + 1 with e ⩾ 1 and an odd prime number ℓ. For 1 ⩽ e ⩽ 4, we prove that h−p is odd whenever 2 is a primitive root modulo ℓ with the help of computer. Without the assumption on ℓ, we compute and find that, in the range e ⩾ 1, ℓ < 220 and p < 237, h−p is even only for four exceptional cases, where ℓ happened to be a Mersenne prime number. Further, computing for larger p with a Mersenne prime ℓ, we find one more exception.

Benchmark Problems for Numerical Schemes to Passively Transported Interface

The interface passively transported by a prescribed flow field is considered, which relates to a part of a mathematical model for two-fluid flow. A benchmark problem is proposed. The shape of the interface transported by the flow field in the problem is deformed but analytically computable. One can fully validate the accuracy of this numerical scheme by the benchmark problem. As an example, computations are done using a scheme for which an interface convergence theorem had been proved. The result does not only show optimality of the theorem but also shows effectiveness of the benchmark problem.