对流环流域上非线性磁热耦合界面问题的浸入非协调有限元方法研究

Magneto-heating coupling interface model is derived from the practical problems such as potential transformer, nuclear fusion and so on, which is very important science meaning to research its theory and numerical analysis. For this program, firstly, by employing the monotone theory or weak star compact imbedding theory, the solvability and stability of the smoothing solutions will be set up. Furthermore, the relationship between the smoothing solutions and original solutions will be established. Then, the solvability and dissipation of the original will be presented. Secondly, the interface can be founded by the level set method or the pixel-scan method for the optimization problem of edge detection, which has to be according with the data structure of the nonconforming finite element methods. Thirdly, the immersed H(curl) and H^1 nonconforming finite element spaces have to be derived with the properties of interpolation estimates, inverse inequalities, trace theory and the estimates of the nonconforming terms. Fourthermore, the discrete schemes will be designed based on the dissipation analysis and the error estimates will be explored as well as the effects of penalty terms are plotted. At last, the software of the numerical experiments will be formed while the compared data and figures will be presented. The aims of the program will be to give the efficient stable theory and numerical approximation methods for magneto-heating coupling interface model for developing and deepening the applications of nonconforming finite element methods.

磁热耦合界面问题是高能变压器、核聚变等现象中抽象产生的数学模型，对它的理论和数值方法进行研究具有重要科学意义。本项目首先分别使用非线性单调算子理论和弱星收敛紧嵌入理论，建立磨光解的可解性分析及磨光解与原问题解之间的收敛关系，并给出原问题的可解性分析、稳定性分析以及耗散分析。其次使用Level Set方法或基于优化问题的像素扫描方法建立界面检测技术，并将界面的数据信息与非协调元的数据结构进行统一，以便进行数值模拟。同时构建界面侵入的H(curl)非协调元和H^1空间非协调元，探讨插值逼近估计、逆不等式、迹估计和非协调项估计等有限元基本性质。进一步构建基于耗散分析的数值离散格式和误差分析，分析惩罚项对收敛的影响。最后通过与其他数值方法进行对比，建立优势评比指标，形成有限元软件包。本项目预期将为高能非线性磁热耦合界面问题提供高效稳定的理论和数值方法，为非协调有限元的发展开拓更加广阔的前景。

磁热耦合界面问题是高能变压器、核聚变等现象中抽象产生的数学模型，对它的理论和数.值方法进行研究具有重要科学意义。本项目首先分别使用非线性单调算子理论和弱星收敛紧嵌.入理论，建立磨光解的可解性分析及磨光解与原问题解之间的收敛关系，并给出原问题的可解.性分析、稳定性分析以及耗散分析。其次使用Level Set方法或基于优化问题的像素扫描方法.建立界面检测技术，并将界面的数据信息与非协调元的数据结构进行统一，以便进行数值模拟.。同时构建界面侵入的H(curl)非协调元和H^1空间非协调元，探讨插值逼近估计、逆不等式、.迹估计和非协调项估计等有限元基本性质。进一步构建基于耗散分析的数值离散格式和误差分.析，分析惩罚项对收敛的影响。最后通过与其他数值方法进行对比，建立优势评比指标，形成.有限元软件包。本项目预期将为高能非线性磁热耦合界面问题提供高效稳定的理论和数值方法.，为非协调有限元的发展开拓更加广阔的前景。

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