Numerical Analysis and Computational Geometry Research of Characteristic Finite Element Methods

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1. Research in Numerical analysisIn characteristic finite element schemes, it is required to integrate functions composed from characteristic mappings and base functions of finite element spaces. Although the integrands are not smooth on each element, it has been experienced that the use of higher-order numerical integration formulas gives better finite element solutions. Its reason and suitable order of the numerical integration formula are studied. A model integration problem on stochastic triangles is introduced. Numerical integration formula that makes the mean error zero is investigated. As a result, a conjecture that provides an appropriate order of numerical integration for the characteristic finite element schemes is obtained, that is, (2k+d)-th order formula is reasonable for the characteristic schemes using Pk finite element in the d-th dimensional space.2. Research in Computational GeometryIn above numerical integrations, the triangle having an upwind point, which is the ima … More ge of an integration node by the characteristic mapping, has to be found. Problem of this type is called the point location problem. For the point location problem, one has proposed the trapezoidal map method, which constructs a data structure for the problem based on trapezoidal subdivision of the domain. Efficiency of the data structure depends on the order of input data. A preprocessing algorithm in order to construct an efficient search tree by one time execution of the trapezoidal map method is proposed.3. Related researches : (1)Analysis of the traction method, which is effective for practical computation of shape optimization problems, (2) Analysis of order of a finite element, which gives the traction method better accuracy in the finite element schemes, (3) Derivation of 2D limit models of Boussinesq equation, which is limit of 3D equation as the thickness tends to zero, (4) A new method which reconstruct fine flaw image from a blurred image obtained by the eddy current testing (ECT) is proposed. (5) A new algorithm which attain the SINC function method by the Fourier transformation of Helmholtz decomposition. Less