We consider a finite element method for elliptic equation with heterogeneous and possibly high-contrast coefficients based on primal hybrid formulation. A space decomposition as in FETI and BDCC allows a sequential computations of the unknowns through elliptic problems and satisfies equilibrium constraints. One of the resulting problems is non-local but with exponentially decaying solutions, enabling a practical scheme where the basis functions have an extended, but still local, support. We obtain quasi-optimal a priori error estimates for low-contrast problems assuming minimal regularity of the solutions. To also consider the high-contrast case, we propose a variant of our method, enriching the space solution via local eigenvalue problems and obtaining optimal a priori error estimate that mitigates the effect of having coefficients with different magnitudes and again assuming no regularity of the solution. The technique developed is dimensional independent and easy to extend to other problems such as elasticity.

翻译：我们提出一种基于原始混合形式的有限元方法，用于求解具有非均匀且可能高对比系数的椭圆型方程。采用类似FETI和BDCC的空间分解方法，通过椭圆问题实现未知量的顺序求解，并满足平衡约束条件。由此产生的子问题之一虽具有非局部特性，但其解呈指数衰减，这使得我们能够设计一种实用方案：基函数虽具有扩展支撑但仍保持局部性。针对低对比度问题，在解仅满足最小正则性的假设下，我们获得了拟最优的先验误差估计。为处理高对比情形，我们提出该方法的变体：通过局部特征值问题对解空间进行扩充，得到最优先验误差估计，该估计消减了不同量级系数的影响，并且同样无需假设解的正则性。所发展的技术具有维度无关性，易于推广至弹性力学等其他问题。