دانلود رایگان مقاله تحلیل خطای پسینی روش تقویت مخلوط المان محدود برای جریان دارسی

Abstract

We develop an a posteriori error analysis of residual type of a stabilized mixed finite element method for Darcy flow. The stabilized formulation is obtained by adding to the standard dual-mixed approach suitable residual type terms arising from Darcy’s law and the mass conservation equation. We derive sufficient conditions on the stabilization parameters that guarantee that the augmented variational formulation and the corresponding Galerkin scheme are well-posed. Then, we obtain a simple a posteriori error estimator and prove that it is reliable and locally efficient. Finally, we provide several numerical experiments that illustrate the theoretical results and support the use of the corresponding adaptive algorithm in practice.

6. Conclusions

We considered a slight generalization of the method introduced by Masud and Hughes in [1] to the Darcy problem of anisotropic porous media flow. The augmented variational formulation is obtained by adding to the classical dual-mixed variational formulation two weighted residual type terms, that are related with Darcy’s law and the mass conservation equation. We provided sufficient conditions on the stabilization parameters that ensure that the augmented weak formulation is well-posed. Under these same hypotheses, we also proved that the corresponding Galerkin scheme is well-posed and a Cea-type estimate holds whatever finite-dimensional subspaces are used. ´ In particular, we provide a priori error bounds when the fluid velocity is approximated by Raviart–Thomas or Brezzi–Douglas–Marini elements, and the pressure is approximated using continuous piecewise polynomials. We remark that in this case local mass conservation is not guaranteed.