This paper presents a symmetric positive definite (SPD) coupling between the boundary element method (BEM) and the finite element method (FEM) in the framework of the numerical manifold method (NMM) for two-dimensional linear elastic static problems. The BEM subdomain is treated as a single mathematical patch whose local approximation is derived from the displacement boundary integral equation, thereby preserving the nonlocal nature of BEM. The remaining domain is covered by a finite element mesh, with each node defining a patch and the associated shape functions serving as weight functions. Weight functions are defined over the entire mathematical cover, with explicit zero values outside the support of each patch. This global definition ensures that the partition of unity holds everywhere and enables the global displacement approximation to be expressed as a superposition of contributions from all patches. Within this unified framework, the interface between the BEM and FEM subdomains emerges naturally as a transition zone of weight functions, rather than a distinct boundary. Displacement continuity is automatically satisfied through the partition of unity, and traction equilibrium is approximately enforced through the variational formulation. To fully incorporate the coupling formulation into the minimum potential energy framework, the tractions on the BEM patch are eliminated in favor of displacements using the displacement boundary integral equation (BIE). Prescribed tractions on the BEM patch are enforced via a penalty method. The resulting algebraic system is symmetric by construction and remains positive definite when either constant or isoparametric boundary elements are used. This work serves as a proof-of-concept study for the SPD coupling framework with constant elements. Numerical examples demonstrate the accuracy and convergence of the method. The results show that the coupling procedure preserves the intrinsic convergence properties of each subdomain: the BEM part converges at a rate close to unity for displacements and approximately 2.0 for stresses, while the FEM part achieves quadratic convergence for both. The study also reveals that near-singular integrals in the strain BIE can affect the convergence rate when the element size becomes sufficiently small.
Zhou et al. (Thu,) studied this question.