Study on non-standard finite element approximation methods for partial differential equations

对于非标准有限元法及相关问题，得到了以下结果：首席研究员T. Kako研究了非局部Dirichlet - Neumann映射的非标准有限元近似方法j在I外无界区域内的传播，并开发了一种有效的高波数二维辐射问题的方法。作为该方法的一个应用，已经可以计算出接近真实元音的形成峰曲线。他发展了声场与壳类结构耦合问题的数值计算方法。他还明确指出了在其FEM中操作人员基本频谱的重要性。以下是调查人员的结果。T. Ushijima研究了二维约化波问题的基本求解方法，得到了收敛性和误差估计。K. Houlka开发了Freeform+项目。H. Kawakawa研究了海岸上的石油附着和渗透现象，得到了其数学模型，并进行了各种应用的数值模拟。Kikuchi通过几个数值算例开发了一种新的板弯曲有限元方法，并研究了Nedelec边单元的效率。D. Koyama结合虚拟域法和Schwarz替代法研究了三维外Helmholtz问题的有限元方法。T. Takeda和M. Fukuhara开发了偏微分方程的结构化神经网络方法。M. Tabata提出了带误差估计的地幔对流问题的新的有限元格式。s - l。张研究了线性方程的各种共轭梯度型方法。T. Miyoshi发现了一个确定裂纹扩展方向的准则。M. Nakao和N. Yamamoto研究了经过验证的有限元计算数值方法，并通过将问题简化为广义矩阵特征值问题，发展了具有验证性的评价有限元逼近性的方法。少

The following results have been obtained for the non-standard finite element method (FEM) and related problems. The head investigator T. Kako studied the wave propagation in the I exterior unbounded region finding the non-standard finite element approximation method j for a non-local Dirichlet to Neumann mapping, and developed an effective method for 2D radiation problem with high wave number. As an application of the method, it has become possible to compute the formant curve close to the one for the real vowels. He developed the numerical method for the coupling problem between the acoustic field and the structure like shell. He also made clear the importance of the essential Spectrum of the operator in its FEM. The followings are the results by investigators. T. Ushijima studied the fundamental solution method for the 2D reduced wave problem and obtained the convergence and the error estimation. K. Houlka developed the Freeform+ project. H. Kawakawa studied the oil adherence and pen … More etration phenomena on the seashore and obtained its mathematical model and did numerical simulations with various applications.F. Kikuchi developed a new FEM for the plate-bending problem with several numerical examples and studied the efficiency of the Nedelec edge element. D. Koyama studied FEM for the 3D exterior Helmholtz problem combining the fictitious domain method and the Schwarz alternative method. T. Takeda and M. Fukuhara developed the structured neural network method for partial differential equations. M. Tabata found the new FEM scheme for earth mantle convection problem with error estimation. S. -L. Zhang studied various conjugate gradient type method for linear equations. T. Miyoshi found a criterion to determine the direction of a crack extension. M. Nakao and N. Yamamoto studied the validated numerical method for FEM computation and developed the method to evaluate the approximation property of FEM with validation by reducing the problem to the generalized matrix eigenvalue problem. Less