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Nonconforming Least-Squares Method for Elliptic Partial Differential Equations with Smooth Interfaces

机译：具有光滑界面的椭圆型偏微分方程的非协调最小二乘方法

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摘要

In this paper a least-squares based method is proposed for elliptic interface problems in two dimensions, where the interface is smooth. The underlying method is spectral element method. The least-squares formulation is based on the minimization of a functional as defined in (4.1). The jump in the solution and its normal derivative across the interface are enforced (in an appropriate Sobolev norm) in the functional. The solution is obtained by solving the normal equations using preconditioned conjugate gradient method. Essentially the method is nonconforming, so a block diagonal matrix is constructed as a preconditioner based on the stability estimate where each diagonal block is decoupled. A conforming solution is obtained by making a set of corrections to the nonconforming solution as in Schwab (p and h-p Finite Element Methods, Clarendon Press, Oxford, 1998) and an error estimate in H~1 -norm is given which shows the exponential convergence of the proposed method.
机译：在本文中，提出了一种基于最小二乘的方法来解决二维光滑的椭圆界面问题。基本方法是光谱元素法。最小二乘公式是基于（4.1）中定义的函数的最小化。在功能中强制执行解决方案中的跳转及其在界面上的正态导数（按照适当的Sobolev规范）。该解决方案是通过使用预处理的共轭梯度法求解法线方程获得的。本质上，该方法是不合格的，因此，基于对角线块解耦的稳定性估计，将块对角线矩阵构造为预处理器。通过对Schwab中的不合格解决方案进行一组校正，可以得到一个合格的解决方案（p和hp Finite Element Methods，Clarendon Press，牛津，1998年），并且给出了H〜1范数的误差估计，该估计表明了指数收敛建议的方法。

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