关于高阶有限元在扩散型方程最大值原理和非负约束方面的性能
On the performance of high-order finite elements with respect to maximum principles and the non-negative constraint for diffusion-type equations
AI总结：
本文通过p-细化（使用非均匀拉格朗日多项式，p=1,...,10）研究稳态各向异性扩散问题，发现p-细化不能消除非负约束的违反，并警示高阶近似用户注意其性能。
摘要：
本文的主要目的是记录p-细化在最大值原理和非负约束方面的性能。模型问题是带有衰减的（稳态）各向异性扩散（这是一个二阶椭圆型偏微分方程）。我们考虑了标准的单场公式（基于Galerkin形式）和两种基于最小二乘的混合公式。我们采用非均匀拉格朗日多项式来改变每个单元的多项式阶数，并使用p=1,...,10。结果表明，对于各向异性扩散，非负约束的违反不会随着p-细化而消失。我们将通过几个代表性例子说明p-细化的性能。本文的预期结果有两方面。首先，本研究将警示高阶近似用户关于其在最大值原理和非负约束方面的性能。其次，本研究将帮助研究人员开发在高阶近似下强制执行最大值原理和非负约束的新方法。
英文摘要：
The main aim of this paper is to document the performance of $p$-refinement with respect to maximum principles and the non-negative constraint. The model problem is (steady-state) anisotropic diffusion with decay (which is a second-order elliptic partial differential equation). We considered the standard single-field formulation (which is based on the Galerkin formalism) and two least-squares-based mixed formulations. We have employed non-uniform Lagrange polynomials for altering the polynomial order in each element, and we have used $p = 1, ..., 10$. It will be shown that the violation of the non-negative constraint will not vanish with $p$-refinement for anisotropic diffusion. We shall illustrate the performance of $p$-refinement using several representative problems. The intended outcome of the paper is twofold. Firstly, this study will caution the users of high-order approximations about its performance with respect to maximum principles and the non-negative constraint. Secondly, this study will help researchers to develop new methodologies for enforcing maximum principles and the non-negative constraint under high-order approximations.
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