Abstract:

As is well known, the study of nonuniform grids can effectively solve the initial value singularity phenomenon of fractional Caputo -type derivatives. In the theoretical analysis of nonuniform grids, fractional discrete Grönwall inequality is often used for error analysis, but there is a lack of specific research on error structures. An error convolution structure (ECS) was designed on nonuniform grids for analyzing the time fractional diffusion wave equation. A quadratic interpolation approximation was applied Caputo -type derivatives, and the BDF2 -type finite element method on nonuniform grids was obtained by discretizing it using a reduction method and a discrete complementary convolution kernel. The discrete complementary convolution kernel is crucial in the convergence analysis of algorithms, as it simplify the process of finite element theory analysis and construct global consistency errors based on the properties of convolution kernels and interpolation estimates. The $L^2$-norm error and $H^1$-norm error of the BDF2 finite element scheme on nonuniform grids were estimated in detail, and verifies the consistency between the proposed finite element scheme and the theoretical convergence order through experiments.

Key words: time fractional diffusion-wave equation, discrete complementary convolution kernels, BDF2 finite element format, error convolutional structure, nonuniform grids