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In this article, an H-Galerkin mixed finite element (MFE) method for solving the time fractional water wave model is presented. First-order backward Euler difference method and L1 formula are applied to approximate integer derivative and Caputo fractional derivative with order 1/2, respectively, and H-Galerkin mixed finite element method is used to approximate the spatial direction. The analysis of stability for fully discrete mixed finite element scheme is made and the optimal space-time orders of convergence for two unknown variables in both H-norm and L-norm are derived. Further, some computing results for a priori analysis and numerical figures based on four changed parameters in the studied problem are given to illustrate the effectiveness of the current method.

FINITE DIFFERENCE/<i>H</i><sup>1</sup>-GALERKIN MFE PROCEDURE FOR A FRACTIONAL WATER WAVE MODEL