论文标题
基于稳态迭代的半线性扩散 - 反应边界值问题的数值最小化方法
A numerical energy minimisation approach for semilinear diffusion-reaction boundary value problems based on steady state iterations
论文作者
论文摘要
我们提出了一个新型的基于能量的数值分析，对半线性扩散 - 反应边界值问题。基于合适的变异设置，建议的计算方案可以看作是一种能量最小化方法。更具体地说，该过程旨在生成一系列数值近似值，这是由相关（稳定）线性化离散问题的迭代解决方案引起的，并且倾向于局部最小值基础能量功能。同时，有限维近似空间是自适应的。这是根据有限元离散化的新的网格完善策略来实现的，这再次依赖于所考虑的问题的能量结构，并且不涉及任何后验错误指标。结合起来，所得的自适应算法由一系列分层精制离散空间的迭代线性化过程组成，我们证明在适当意义上将其收敛于连续问题的解决方案。数值实验证明了我们方法在一系列示例中的鲁棒性和可靠性。
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.
