Adaptive cell method with constitutive matrix correction for simulating physical field on a coarse grid

Highlights: • Calculation error originates from the approximation of constitutive equations. • Universal constitutive matrices applied to the Cartesian and curvilinear grids. • Improve accuracy significantly without changing the grid structure and stencil size. • A general framework to handle a wide variety of applications. -- Abstract: An adaptive cell method is proposed in order to simulate physical fields on a coarse grid with high accuracy. Compared with the conventional cell method, the advance of the proposed adaptive method originates from the so-called node weight correction procedure by using a patch-recovery technique, which corrects the local constitutive relations and in turn mitigates the error source that results from the discretization of constitutive equations. Furthermore, the proposed method has been extended to general curvilinear grids, in which the construction of the local constitutive matrix is derived by using differential geometry. Finally, we have demonstrated by theoretical analysis and numerical results that the proposed strategies can achieve a high-order convergence as well as high-efficiency.