No abstract

In this paper, two efficient two-grid algorithms for the convection-diffusion problem with a modified characteristic finite element method are studied. We present an optimal error estimate in L p -norm for the characteristic finite element method unconditionally, while all previous works require certain time-step restrictions. To linearize the characteristic method equations, two-grid algorithms based on the Newton iteration approach and the correction method are applied. The error estimate and the convergence result of the two-grid method are derived in detail. It is shown that the coarse space can be extremely coarse and achieve asymptotically optimal approximations as long as the mesh sizes H = O h 1 / 3 in the first algorithm and H = O h 1 / 4 in the second algorithm, respectively. Finally, two numerical examples are presented to demonstrate the theoretical analysis.

In this paper, the full discrete scheme of mixed finite element approximation is introduced for semilinear hyperbolic equations. To solve the nonlinear problem efficiently, two two-grid algorithms are developed and analyzed. In this approach, the nonlinear system is solved on a coarse mesh with width H, and the linear system is solved on a fine mesh with width h ≪ H. Error estimates and convergence results of two-grid method are derived in detail. It is shown that if we choose H = (h 1 3 ) in the first algorithm and H = (h 1 4 ) in the second algorithm, the two-grid algorithms can achieve the same accuracy of the mixed finite element solutions. Finally, the numerical examples also show that the two-grid method is much more efficient than solving the nonlinear mixed finite element system directly. KEYWORDSerror estimate, hyperbolic equations, mixed finite element method, two-grid method and boundary conditionwhere Ω ⊂ R 2 is a bounded polygonal domain, is the unit exterior normal to Ω, J = (0, T], K ∶ Ω × R → R 2×2 is a symmetric and uniformly positive definite matrix, u tt and u t denote 2 u t 2 and u t , respectively. Let p = −K∇u, so that Equation 1 may be rewritten as follows: u tt + ∇ · p = (u), (x, t) ∈ Ω × (0, T], K −1 p + ∇u = 0, (x, t) ∈ Ω × (0, T].(4) 3370