A gradient recovery method based on an oblique projection and boundary modification

The gradient recovery method is a technique to improve the approximation of the gradient of a solution by using post-processing methods. We use an $L^2$-projection based on an oblique projection, where the trial and test spaces differ, for efficient numerical computation. We modify our oblique projection by applying the boundary modification method to obtain higher order approximation on the boundary patch. Numerical examples are presented to demonstrate the efficiency and optimality of the approach. References H. Guo, Z. Zhang, R. Zhao and Q. Zou. Polynomial preserving recovery on boundary. J. Comput. Appl. Math., 307:119–133, 2016. doi:10.1016/j.cam.2016.03.003 M. Ilyas and B. P. Lamichhane. A stabilised mixed finite element method for the Poisson problem based on a three-field formulation. In editors M. Nelson, D. Mallet, B. Pincombe and J. Bunder Proceedings of the 12th Biennial Engineering Mathematics and Applications Conference, EMAC-2015, volume 57 of ANZIAM J., pages C177–C192, September 2016. doi:10.21914/anziamj.v57i0.10356 M. Krizek and P. Neittaanmaki. Superconvergence phenomenon in the finite element method arising from averaging gradients. Numerische Mathematik, 45(1):105–116, 1984. doi:10.1007/BF01379664 B. P. Lamichhane. A gradient recovery operator based on an oblique projection. Electron. Trans. Numer. Anal., 37:166–172, 2010. http://emis.ams.org/journals/ETNA/vol.37.2010/pp166-172.dir/pp166-172.pdf B. P. Lamichhane. Mixed finite element methods for the Poisson equation using biorthogonal and quasi-biorthogonal systems. Advances in Numerical Analysis, 2013:189045, 2013. doi:10.1155/2013/189045 B. P. Lamichhane. A finite element method for a biharmonic equation based on gradient recovery operators. BIT Numerical Mathematics, 54(2):469–484, 2014. doi:10.1007/s10543-013-0462-0 B. P. Lamichhane, R. P. Stevenson and B. I. Wohlmuth. Higher order mortar finite element methods in 3D with dual Lagrange multiplier bases. Numerische Mathematik, 102(1):93–121, 2005. doi:10.1007/s00211-005-0636-z J. Xu and Z. Zhang. Analysis of recovery type a posteriori error estimators for mildly structured grids. Math. Comp., 73(247):1139–1152, 2004. doi:10.1090/S0025-5718-03-01600-4 Z. Zhang and A. Naga. A new finite element gradient recovery method: superconvergence property. SIAM J. Sci. Comput., 26(4):1192–1213, 2005. doi:10.1137/S1064827503402837 O. C. Zienkiewicz and J. Z. Zhu. The superconvergent patch recovery and a posteriori error estimates. I. The recovery technique. Internat. J. Numer. Methods Engrg., 33(7):1331–1364, 1992. doi:10.1002/nme.162033070

A C0C^0 Linear Finite Element Method for a Second Order Elliptic Equation in Non-Divergence Form with Cordes Coefficients

In this paper, we develop a gradient recovery based linear (GRBL) finite element method (FEM) and a Hessian recovery based linear (HRBL) FEM for second order elliptic equations in non-divergence form. The elliptic equation is casted into a symmetric non-divergence weak formulation, in which second order derivatives of the unknown function are involved. We use gradient and Hessian recovery operators to calculate the second order derivatives of linear finite element approximations. Although, thanks to low degrees of freedom (DOF) of linear elements, the implementation of the proposed schemes is easy and straightforward, the performances of the methods are competitive. The unique solvability and the $H^2$ seminorm error estimate of the GRBL scheme are rigorously proved. Optimal error estimates in both the $L^2$ norm and the $H^1$ seminorm have been proved when the coefficient is diagonal, which have been confirmed by numerical experiments. Superconvergence in errors has also been observed. Moreover, our methods can handle computational domains with curved boundaries without loss of accuracy from approximation of boundaries. Finally, the proposed numerical methods have been successfully applied to solve fully nonlinear Monge-Amp\`{e}re equations

Polynomial Preserving Recovery For Weak Galerkin Methods And Their Applications

Gradient recovery technique is widely used to reconstruct a better numerical gradient from a finite element solution, for mesh smoothing, a posteriori error estimate and adaptive finite element methods. The PPR technique generates a higher order approximation of the gradient on a patch of mesh elements around each mesh vertex. It can be used for different finite element methods for different problems. This dissertation presents recovery techniques for the weak Galerkin methods and as well as applications of gradient recovery on various of problems, including elliptic problems, interface problems, and Stokes problems. Our first target is to develop a boundary strategy for the current PPR algorithm. The current accuracy of PPR near boundaries is not as good as that in the interior of the domain. It might be even worse than without recovery. Some special treatments are needed to improve the accuracy of PPR on the boundary. In this thesis, we present two boundary recovery strategies to resolve the problem caused by boundaries. Numerical experiments indicate that both of the newly proposed strategies made an improvement to the original PPR. Our second target is to generalize PPR to the weak Galerkin methods. Different from the standard finite element methods, the weak Galerkin methods use a different set of degrees of freedom. Instead of the weak gradient information, we are able to obtain the recovered gradient information for the numerical solution in the generalization of PPR. In the PPR process, we are also able to recover the function value at the nodal points which will produce a global continuous solution instead of piecewise continuous function. Our third target is to apply our proposed strategy and WGPPR to interface problems. We treat an interface as a boundary when performing gradient recovery, and the jump condition on the interface can be well captured by the function recovery process. In addition, adaptive methods based on WGPPR recovery type a posteriori error estimator is proposed and numerically tested in this thesis. Application on the elliptic problem and interface problem validate the effectiveness and robustness of our algorithm. Furthermore, WGPPR has been applied to 3D problem and Stokes problem as well. Superconvergent phenomenon is again observed