This paper presents a numerical comparison of finite-element methods resulting in local mass conservation at the element level for Poisson's problem, namely the primal hybrid and mixed methods. These formulations result in an indefinite system. Alternative formulations yielding a positive-definite system are obtained after hybridizing each method. The choice of approximation spaces yields methods with enhanced accuracy for the pressure variable, and results in systems with identical size and structure after static condensation. A regular pressure precision mixed formulation is also considered based on the classical RT space. The simulations are accelerated using Open multi-processing (OMP) and Thread-Building Blocks (TBB) multithreading paradigms alongside either a coloring strategy or atomic operations ensuring a thread-safe execution. An additional parallel strategy is developed using C++ threads, which is based on the producer-consumer paradigm, and uses locks and semaphores as synchronization primitives. Numerical tests show the optimal parallel strategy for these finite-element formulations, and the computational performance of the methods are compared in terms of simulation time and approximation errors. Additional results are developed during the process. Numerical solvers often fail to find an accurate solution to the highly indefinite systems arising from finite-element formulations, and this paper documents a matrix ordering strategy to stabilize the resolution. A procedure to enable static condensation based on the introduction of piecewise constant functions that also fulfills Neumann's compatibility condition, and yet computes an average pressure per element is presented.

中文翻译：

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导致泊松问题质量守恒的有限元方法的比较数值研究：原始混合、混合及其混合公式


本文对泊松问题的单元级局部质量守恒有限元方法（即原始混合方法和混合方法）进行了数值比较。这些公式导致了一个不定系统。在混合每种方法后，获得了产生正定系统的替代配方。近似空间的选择产生了压力变量精度更高的方法，并导致静态冷凝后具有相同尺寸和结构的系统。基于经典RT空间还考虑了常规压力精确混合公式。使用开放式多处理 (OMP) 和线程构建块 (TBB) 多线程范例以及确保线程安全执行的着色策略或原子操作来加速模拟。另一种并行策略是使用 C++ 线程开发的，该策略基于生产者-消费者范例，并使用锁和信号量作为同步原语。数值测试显示了这些有限元公式的最佳并行策略，并在模拟时间和近似误差方面比较了这些方法的计算性能。在此过程中会产生更多结果。数值求解器通常无法找到由有限元公式产生的高度不定系统的准确解，本文记录了一种稳定分辨率的矩阵排序策略。提出了一种基于引入分段常数函数实现静态凝结的过程，该函数也满足诺伊曼的兼容性条件，并且计算每个单元的平均压力。