Study of methodology for a posterior error estimation of finite element solutions

作为对有限元方法后验误差估计方法和技术的研究，我们获得了以下结果和观察。基于超圆法的后验误差估计研究：对于泊松方程，我们用混合有限元方法和相关方法对上述后验方法进行了分析，并通过数值实验对结果进行了验证。一些研究结果发表在一份国际期刊上。结果表明，本方法所需误差常数很少，但适用范围有限。我们还研究了它在更广泛的问题和不符合FEM中的有效性。插值误差常数的估计：分析了常、协调线性三角形有限元中出现的各种插补误差常数。在后验误差估计和Nakao的数值验证方法中，这些常数的定量评估是必不可少的。研究结果发表在国际期刊上。对非协调线性三角形也进行了类似的分析。电磁学有限元的发展与分析：关于第1、2项，我们一直在研究电磁学有限元。关于四边形单元的一些结果作为一项国际联合工作发表在一份国际期刊上。我们用傅立叶有限元分析了轴对称区域上的麦克斯韦本征值问题。板弯曲有限元研究：我们研究并在国际期刊上发表了一种统一的Kirchhoff单元和Reissner-Mindlin单元的方法。我们正在研究横向剪力的确定方法。间断Galerkin方法的研究：我们开始了间断Galerkin方法的后验误差分析。第二项中的非协调有限元就是这种方法的一个经典例子，也是我们研究的一个起点。

As a study of methods and techniques for a posteriori error estimation of the finite element method (FEM), we obtained the following results and observations.1. Study of a posteriori error estimation based on the hypercircle method : For Poisson's equation, we have analyzed the above a posteriori method with the mixed FEM and related methods, and verified the results by numerical experiment. Some results were published in an international journal. The present method turns out to require very few error constants but its range of applicability is limited. We also search for its effectiveness in wider problems and non-conforming FEM.2. Estimation of interpolation error constants : We analyzed various interpolation error constants appearing in the constant and conforming linear triangular finite elements. Quantitative evaluation of such constants is essential in a posteriori error estimation and Nakao's numerical verification method. The results were published in international journals. We also perform similar analysis for the non-conforming linear triangle.3. Development and analysis of FEM for electromagnetics : Related to items 1 and 2, we have been studying FEM for electromagnetics. Some results for quadrilateral elements were published in an international journal as an international joint work. We are analyzing Maxwell's eigenvalue problems over axisymmetric domains by the Fourier FEM.4. Study of plate bending FEM : We have studied and published a unification method for the Kirchhoff and Reissner-Mindlin elements in an international journal. We are now studying determination method for the transverse shear forces.5. Study of the discontinuous Galerkin method : We start a posteriori error analysis of the discontinuous Galerkin method. The non-conforming FEM in item 2 is a classical example of such method, and is taken as a starting point of our study.