Analysis of adaptive two-grid finite element algorithms for linear and nonlinear problems

Abstract: This paper proposes some efficient and accurate adaptive two-grid (ATG) finite element algorithms for linear and nonlinear partial differential equations (PDEs). The main idea of these algorithms is to utilize the solutions on the $k$-th level adaptive meshes to find the solutions on the $(k+1)$-th level adaptive meshes which are constructed by performing adaptive element bisections on the $k$-th level adaptive meshes. These algorithms transform non-symmetric positive definite (non-SPD) PDEs (resp., nonlinear PDEs) into symmetric positive definite (SPD) PDEs (resp., linear PDEs). The proposed algorithms are both accurate and efficient due to the following advantages: they do not need to solve the non-symmetric or nonlinear systems; the degrees of freedom (d.o.f.) are very small; they are easily implemented; the interpolation errors are very small. Next, this paper constructs residue-type {\em a posteriori} error estimators, which are shown to be reliable and efficient. The key ingredient in proving the efficiency is to establish an upper bound of the oscillation terms, which may not be higher-order terms (h.o.t.) due to the low regularity of the numerical solution. Furthermore, the convergence of the algorithms is proved when bisection is used for the mesh refinements. Finally, numerical experiments are provided to verify the accuracy and efficiency of the ATG finite element algorithms, compared to regular adaptive finite element algorithms and two-grid finite element algorithms [27].