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

对非标准有限元方法及相关问题进行了研究，得到了以下结果。首席研究员T.Kako研究了波在非局部Dirichlet-Neumann映射的外部无界区域的传播，找到了非局部Dirichlet到Neumann映射的非标准有限元近似方法j，发展了一种求解高波数二维辐射问题的有效方法。作为该方法的一个应用，计算出的共振峰曲线接近于真实元音的共振峰曲线。他提出了声场与壳体结构耦合问题的数值方法。他还明确了算子本质谱在其有限元分析中的重要性。以下是调查人员的调查结果。T.Ushijima研究了二维约化波问题的基本解方法，得到了收敛和误差估计。K.Houlka开发了Free Form+项目。川川方明研究了油膜粘附性和PEN…F.Kikuchi发展了一种新的板弯曲问题的有限元方法，并给出了几个数值算例，研究了Nedelec边单元的效率。D.Koyama结合虚拟域法和Schwarz交替法研究了三维Helmholtz外问题的有限元方法。T.Takeda和M.Fukuhara发展了求解偏微分方程组的结构化神经网络方法。M.Tabata提出了一种新的求解地幔对流问题的有限元格式，并进行了误差估计。S.-L.张学良研究了解线性方程组的各种共轭梯度法。T.Miyoshi发现了一个确定裂纹扩展方向的标准。M.Nakao和N.Yamamoto研究了有限元计算的验证数值方法，并通过将问题归结为广义矩阵特征值问题，发展了用验证来评估有限元逼近性能的方法。较少

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