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

Yukun Li

Published 2018-05-21Version 1

This paper proposes some novel efficient and accurate adaptive two-grid (ATG) finite element algorithms for linear and nonlinear partial differential equations (PDEs). In these algorithms, they use the information of the solutions on k-th level adaptive meshes, instead of on the uniform meshes, to find the solutions on (k+1)-th level adaptive meshes. They transform the non-symmetric positive definite (non-SPD) PDEs into symmetric positive definite (SPD) PDEs, and transform the nonlinear PDEs into the linear PDEs. These algorithms have the following advantages: 1. Comparing with adaptive methods, they do not need to solve the nonlinear systems; 2. Comparing with two-grid methods, the degrees of freedom are largely reduced; 3. Comparing with the cases when uniform meshes are used for coarse level approximation, they are easily implemented; they are more efficient and accurate since only the interpolation of the solution on newly refined meshes needs to be computed, and the interpolation error is also reduced; they are especially efficient when many steps of mesh refinements are used since the computational cost of computing solutions on uniform meshes is large then. Next, this paper constructs a residue-type a posteriori error estimator for general non-SPD linear problems. We prove the upper bound of the oscillation term, and this gets rid of the assumption that the oscillation term is a high order term (h.o.t.), which may not be true generally due to the low regularity of the numerical solution. Based on this result, the reliability and efficiency of the error estimator are established. Finally, the convergence of the error on the adaptive meshes is proved when bisection is used for the mesh refinement.