We introduce a numerical scheme that approximates solutions to linear PDE’s by minimizing a residual in the \(W^{-1,p'}(\Omega )\) norm with exponents \(p> 2\). The resulting problem is solved by regularized Kačanov iterations, allowing to compute the solution to the non-linear minimization problem even for large exponents \(p\gg 2\). Such large exponents remedy instabilities of finite element methods for problems like convection-dominated diffusion.

中文翻译：

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求解具有大指数 p 的 $$W^{-1,p'}$$ 中的最小残差方法

我们引入了一种数值方案，通过最小化指数 \ (p> 2\) 的\(W^{-1,p'}(\Omega )\)范数中的残差来近似线性偏微分方程的解。由此产生的问题通过正则化 Kačanov 迭代来解决，即使对于大指数\(p\gg 2\)也可以计算非线性最小化问题的解决方案。如此大的指数弥补了有限元方法在解决对流主导扩散等问题时的不稳定性。