论文标题
非线性退化抛物线方程的数值方法的均匀收敛性
Uniform-in-time convergence of numerical methods for non-linear degenerate parabolic equations
论文作者
论文摘要
梯度方案是一个框架，可实现椭圆形和抛物线偏微分方程的许多数值方法的统一收敛分析：符合和不合格的有限元，混合有限元和有限体积方法。我们在这里表明，该框架可以应用于一个退化的非线性抛物线方程（尤其是理查兹，Stefan's and Leray-Lions'模型），我们证明了这些方程式的梯度方案近似值近似值。为了建立这种融合，我们为抛物线模型的数值近似值开发了几种离散的紧凑工具，包括不连续的asscoli-arzelà定理和一个均匀的时间弱空间内置离散的aubin-simon定理。该模型的退化性发生在时间和空间衍生物中，也要求我们开发离散的补偿紧凑性结果。
Gradient schemes is a framework that enables the unified convergence analysis of many numerical methods for elliptic and parabolic partial differential equations: conforming and non-conforming Finite Element, Mixed Finite Element and Finite Volume methods. We show here that this framework can be applied to a family of degenerate non-linear parabolic equations (which contain in particular the Richards', Stefan's and Leray--Lions' models), and we prove a uniform-in-time strong-in-space convergence result for the gradient scheme approximations of these equations. In order to establish this convergence, we develop several discrete compactness tools for numerical approximations of parabolic models, including a discontinuous Ascoli-Arzelà theorem and a uniform-in-time weak-in-space discrete Aubin-Simon theorem. The model's degeneracies, which occur both in the time and space derivatives, also requires us to develop a discrete compensated compactness result.
