Takuya Tsuchiya

Finite difference, finite element and finite volume methods applied to two-point boundary value problems

This paper considers the finite difference, finite element and finite volume methods applied to the two-point boundary value problem -d/dx(p(x) du/dx) = f(x), a < x < b, u(a) = u(b) = 0. By using an inversion formula of a nonsingular tridiagonal matrix, explicit expressions of approximate solutions by three methods are given, which lead to a unified understanding of these methods as well as their unified error estimates. Numerical examples are also given.

On the circumradius condition for piecewise linear triangular elements

We discuss the error analysis of linear interpolation on triangular elements. We claim that the circumradius condition is more essential than the well-known maximum angle condition for convergence of the finite element method, especially for the linear Lagrange finite element. Numerical experiments show that this condition is the best possible. We also point out that the circumradius condition is closely related to the definition of surface area.

A Babuška-Aziz type proof of the circumradius condition

In this paper the error of polynomial interpolation of degree 1 on triangles is considered. The circumradius condition, which is more general than the maximum angle condition, is explained and proved by the technique given by Babuška-Aziz.

A priori error estimates of finite element solutions of parametrized strongly nonlinear boundary value problems

Abstract Nonlinear boundary value problems with parameters are called parametrized nonlinear boundary problems. This paper studies a priori error estimates of finite element solutions of second-order parametrized strongly nonlinear boundary value problems in divergence form on one-dimensional bounded intervals. The Banach space W 0 1, ∞ is chosen in formulation of the error analysis so that the nonlinear differential operators defined by the differential equations are nonlinear Fredholm operators of index 1. Finite element solutions are defined in a natural way, and several a priori estimates are proved on regular branches and on branches around turning points. In the proofs the extended implicit function theorem due to Brezzi et al. (1980) plays an essential role.

A priori error estimates for Lagrange interpolation on triangles

We present the error analysis of Lagrange interpolation on triangles. A new a priori error estimate is derived in which the bound is expressed in terms of the diameter and circumradius of a triangle. No geometric conditions on triangles are imposed in order to get this type of error estimates. To derive the new error estimate, we make use of the two key observations. The first is that squeezing a right isosceles triangle perpendicularly does not reduce the approximation property of Lagrange interpolation. An arbitrary triangle is obtained from a squeezed right triangle by a linear transformation. The second key observation is that the ratio of the singular values of the linear transformation is bounded by the circumradius of the target triangle.

Finite element approximations of conformal mappings

In this paper, finite element approximations of conformal mappings defined on the unit disk to Jordan domains are considered. Three types of the normalization conditions are dealt with. In each case, convergence of the finite element conformal mappings to the exact conformal mappings is proved. Several numerical examples are given.

An application of the Kantorovich theoremto nonlinear finite element analysis

Summary. Finite element solutions of strongly nonlinear elliptic boundary value problems are considered. In this paper, using the Kantorovich theorem, we show that, if the Fréchet derivative of the nonlinear operator defined by the boundary value problem is an isomorphism at an exact solution, then there exists a locally unique finite element solution near the exact solution. Moreover, several a priori error estimates are obtained.

Numerical verification of solutions of parametrized nonlinear boundary value problems with turning points

Nonlinear boundary value problems (NBVPs in abbreviation) with parameters are called parametrized nonlinear boundary value problems. This paper studies numerical verification of solutions of parametrized NBVPs defined on one-dimensional bounded intervals. Around turning points the original problem is extended so that the extended problem has an invertible Fréchet derivative. Then, the usual procedure of numerical verification of solutions can be applied to the extended problem. A numerical example is given.

Yamamoto's principle and its applications to precise finite element error analysis

Suppose that we discretize an elliptic boundary value problem and obtain a linear equation Ax = f. In many case, the inverse matrix A-1 is closely related to the Green function of the original boundary value problem. This fact is called Yamamotos principle. In this paper, using Yamamotos principle, we develop a precise error analysis of the piecewise linear finite element method for two-point boundary value problems with discontinuous and not necessarily positive coefficient functions. We show that precise error estimations, similar to known error bounds, are obtained even in this case.

An Explicit Inversion Formula for Tridiagonal Matrices

Discretizing two-point boundary value problems on an interval by finite difference method, we obtain a certain type of tridiagonal coefficient matrices. In this paper we give an explicit inversion formula for such tridiagonal matrices using Yamamoto-Ikebe’s inversion formula.