满足离散最大值原理的时间分数阶对流扩散反应方程的稳定化格式
A stabilized scheme satisfying the discrete maximum principle for a time fractional convection-diffusion-reaction equation
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中文总结 AI 辅助
针对有界域中的时间分数阶对流扩散反应方程，结合协调线性有限元与代数通量校正方法构造满足离散最大值原理的稳定化格式，经理论证明适定性与误差估计，数值实验验证了收敛阶及对层状解的适用性。
中文摘要 AI 辅助
我们研究有界域Ω⊂ℝ²中的时间分数阶对流扩散反应方程。通过将协调线性有限元方法与代数通量校正方法相结合，构造了一种满足离散最大值原理的稳定化数值格式，所得半离散格式为非线性的，且已证明其适定性。在假设初始数据非光滑的情况下，我们利用能量方法推导了半离散格式的误差估计。对于时间离散，采用L1方法得到全离散格式，我们证明了该格式的适定性和离散最大值原理。我们还给出了数值实验，验证了其收敛阶，并对具有层状结构的解测试了所构造的格式。
英文摘要
We study a time fractional convection-diffusion-reaction equation in a bounded domain $Ω\subset\mathbb{R}^2$. A stabilized numerical scheme satisfying a discrete maximum principle is constructed by combining the conforming linear finite element method with the algebraic flux correction method. The resulting semi-discrete scheme is nonlinear, and its well-posedness is established. Assuming nonsmooth initial data, we derive error estimates for the semi-discrete scheme using energy arguments. For the temporal discretization, we employ the L1 method, obtaining a fully discrete scheme for which we prove well-posedness and the discrete maximum principle. We also present numerical experiments that validate the order of convergence as well as we test our schemes to solutions that possess layers.
发表机构
- Friedrich-Schiller-Universität Jena（耶拿弗里德里希·席勒大学）
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