论文标题
普遍的FEM的指数收敛构成异质反应扩散方程
Exponential convergence of a generalized FEM for heterogeneous reaction-diffusion equations
论文作者
论文摘要
提出了一种广义有限元方法，用于求解具有单数扰动参数$ \ varepsilon $的异质反应扩散方程，该方程是基于局部通过局部反应 - 延伸方程和A局部特征性特征性的局部反应 - 延伸方程的溶液近似于每个子域上的溶液。这些局部问题在某些域中略大于少量采样大小$δ^{\ ast} $的子域。该方法是在连续级别的连续性离散化和离散级别的直接离散化，作为其标准FE离散化的粗空近似。相对于$δ^{\ ast}/\ varepsilon $和$δ^{\ ast}/h $的局部近似错误的指数衰减率（在离散级别上，用$ h $表示罚款Fe Mesh尺寸）并建立了本地自由度。特别是，这表明，连续级别的方法相对于$ \ varepsilon $在标准$ h^{1} $规范的标准中均匀收敛，如果相对于$ \ varepsilon $ and $ h $（在离职的级别）相对于局部反应范围的解决方案，则对局部反应的解决方案和近似值的解决方案相对于$ \ varepsilon $ and $ h $相对较大。提供数值结果以验证理论结果。
A generalized finite element method is proposed for solving a heterogeneous reaction-diffusion equation with a singular perturbation parameter $\varepsilon$, based on locally approximating the solution on each subdomain by solution of a local reaction-diffusion equation and eigenfunctions of a local eigenproblem. These local problems are posed on some domains slightly larger than the subdomains with oversampling size $δ^{\ast}$. The method is formulated at the continuous level as a direct discretization of the continuous problem and at the discrete level as a coarse-space approximation for its standard FE discretizations. Exponential decay rates for local approximation errors with respect to $δ^{\ast}/\varepsilon$ and $δ^{\ast}/h$ (at the discrete level with $h$ denoting the fine FE mesh size) and with the local degrees of freedom are established. In particular, it is shown that the method at the continuous level converges uniformly with respect to $\varepsilon$ in the standard $H^{1}$ norm, and that if the oversampling size is relatively large with respect to $\varepsilon$ and $h$ (at the discrete level), the solutions of the local reaction-diffusion equations provide good local approximations for the solution and thus the local eigenfunctions are not needed. Numerical results are provided to verify the theoretical results.
