Coupled multi-physics problems are encountered in countless applications and pose significant numerical challenges. Although monolithic approaches offer possibly the best solution strategy, they often require ad-hoc preconditioners and numerical implementations. Sequential (also known as splitted, partitioned or segregated) approaches are iterative methods for solving coupled problems where each equation is solved independently and the coupling is achieved through iterations. These methods offer the possibility to flexibly add or remove equations from a model and to rely on existing black-box solvers for every specific equation. Furthermore, when problems are non-linear, inner iterations need to be performed even in monolithic solvers, therefore making a sequential iterative approach a viable alternative. The cost of running inner iterations to achieve the coupling, however, could easily becomes prohibitive, or, in some cases the iterations might not converge. In this work we present a general formulation of splitting schemes for continuous operators, with arbitrary implicit/explicit splitting, like in standard iterative methods for linear systems. By introducing a generic relaxation operator we find the conditions for the convergence of the iterative schemes. We show how the relaxation operator can be thought as a preconditioner and constructed based on an approximate Schur-complement. We propose a Schur-based Partial Jacobi relaxation operator to stabilise the coupling and show its effectiveness. Although we mainly focus on scalar-scalar linear problems, most results are easily extended to non-linear and higher-dimensional problems. Numerical tests (1D and 2D) for two PDE systems, namely the Dual-Porosity model and a Quad-Laplacian operator, are carried out to confirm the theoretical results.

翻译：耦合多物理场问题在众多应用中普遍存在，并带来显著的数值挑战。尽管整体式方法可能提供最好的求解策略，但它们通常需要专门预处理器和数值实现。顺序方法（亦称分裂、分区或解耦方法）是求解耦合问题的迭代方法，其中每个方程独立求解，通过迭代实现耦合。这些方法能够灵活地向模型添加或移除方程，并可依赖针对特定方程的现成黑箱求解器。此外，当问题非线性时，即使整体式求解器也需要执行内迭代，因此顺序迭代方法成为可行的替代方案。然而，实现耦合所需的内迭代成本可能变得过高，或迭代可能不收敛。本文提出连续算子分裂格式的一般性公式，允许任意隐式/显式分裂（类似线性系统标准迭代方法）。通过引入通用松弛算子，我们找到了迭代格式收敛的条件。我们展示了如何将松弛算子视为预处理算子，并基于近似Schur补构造该算子。我们提出一种基于Schur的部分Jacobi松弛算子以稳定耦合，并验证其有效性。尽管主要关注标量-标量线性问题，多数结果可轻松推广至非线性和更高维问题。通过对两个偏微分方程组系统（即双孔隙模型和四阶拉普拉斯算子）进行数值测试（一维和二维），验证了理论结果。