基于稳态迭代的半线性扩散-反应边值问题能量最小化方法
A numerical energy minimisation approach for semilinear diffusion-reaction boundary value problems based on steady state iterations
摘要：
提出了一种新的基于能量的半线性扩散-反应边值问题的数值分析方法。基于一个合适的变分设置，所提出的计算方案可以被看作是一个能量最小化的方法。更具体地说，这个过程的目的是产生一个序列的数值逼近，这是从相关的（稳定）线性化离散问题的迭代解，并趋于一个局部极小值的基本能量功能。同时，有限维近似空间的自适应细化，这是实现在有限元离散化的背景下，这再次依赖于能量结构的问题，并不涉及任何后验误差指标的一个新的网格细化策略。在组合中，所得到的自适应算法包括一个迭代线性化过程的序列分层细化离散空间，我们证明收敛到一个解决方案的连续问题在适当的意义上。数值实验表明，我们的方法的鲁棒性和可靠性的一系列例子。
英文摘要：
We present a novel energy-based numerical analysis of semilinear diffusion-reaction boundary value problems. Based on a suitable variational setting, the proposed computational scheme can be seen as an energy minimisation approach. More specifically, this procedure aims to generate a sequence of numerical approximations, which results from the iterative solution of related (stabilised) linearised discrete problems, and tends to a local minimum of the underlying energy functional. Simultaneously, the finite-dimensional approximation spaces are adaptively refined; this is implemented in terms of a new mesh refinement strategy in the context of finite element discretisations, which again relies on the energy structure of the problem under consideration, and does not involve any a posteriori error indicators. In combination, the resulting adaptive algorithm consists of an iterative linearisation procedure on a sequence of hierarchically refined discrete spaces, which we prove to converge towards a solution of the continuous problem in an appropriate sense. Numerical experiments demonstrate the robustness and reliability of our approach for a series of examples.
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