Two-level methods for solving higher order finite element discretizations of nonsymmetric and indefinite elliptic problem

Several two-level methods are proposed to solve the higher-order finite element discretizations of nonsymmetric and indefinite problem. Compared with the two-grid algorithm, the two-level algorithms advanced in this paper only need a fixed grid with mesh size H. The first two-level method is to solve the original problem using a lower-order finite element space \(V^s_H(1\leqslant s\leqslant l)\) $V_H^s(1⩽s⩽l)$ with integers s and l, and then solve a corresponding symmetric positive definite problem using a higher-order finite element space \(V^{l+1}_H\) $V_H^{l+1}$ . The corresponding convergence order is \(O(H^{s+1})\) $O(H^{s+1})$ which can be equal to one of two-grid algorithms. The second two-level algorithm is to add the correction on \(V_H^t(1 \leqslant t \leqslant l)\) $V_H^t(1⩽t⩽l)$ with integer t to the first algorithm, and the corresponding \(L^{2}\) $L^2$ error estimate is one order higher than that of the first algorithm but independent of t. At last, numerical experiments are presented to demonstrate the effectiveness of two-level algorithms, less CPU time is consumed to achieve the same level of accuracy compared with two-grid algorithm.