To ensure preservation of local or global bounds for numerical solutions of conservation laws, we constrain a baseline finite element discretization using optimization-based (OB) flux correction. The main novelty of the proposed methodology lies in the use of flux potentials as control variables and targets of inequality-constrained optimization problems for numerical fluxes. In contrast to optimal control via general source terms, the discrete conservation property of flux-corrected finite element approximations is guaranteed without the need to impose additional equality constraints. Since the number of flux potentials is less than the number of fluxes in the multidimensional case, the potential-based version of optimal flux control involves fewer unknowns than direct calculation of optimal fluxes. We show that the feasible set of a potential-state potential-target (PP) optimization problem is nonempty and choose a primal-dual Newton method for calculating the optimal flux potentials. The results of numerical studies for linear advection and anisotropic diffusion problems in 2D demonstrate the superiority of the new OB-PP algorithms to closed-form flux limiting under worst-case assumptions.
翻译:为确保保护法律数字解决办法的当地或全球界限的保全,我们采用优化(OB)通量校正,限制使用基准有限要素的离散。拟议方法的主要新颖之处在于将通量潜能作为控制变量和因不平等而受限制的对数字通量的优化问题的目标。与以一般来源术语进行最佳控制相比,通量校正的有限要素近似离散保护特性得到保障,而不必施加额外的平等限制。由于通量潜力的数量少于多层面情况中的通量,最佳通量控制的潜在版本的未知数比最佳通量的直接计算要少。我们表明,一套可行的潜在状态潜在目标优化问题并非空闲,而是选择一种初步的牛顿方法来计算最佳通量潜力。2D中线性静态和异性扩散问题的数字研究结果表明,在最坏假设下,新OB-P值算法与封闭式通量的通量具有优势。