A new mixed element method for a class of time-fractional partial differential equations

Area of Science:

• Numerical Analysis
• Computational Mathematics
• Partial Differential Equations

Background:

• Time-fractional partial differential equations (PDEs) are crucial in modeling complex phenomena.
• Existing numerical methods often involve complex mathematical spaces, posing computational challenges.

Purpose of the Study:

• To introduce a new mixed element method for solving time-fractional PDEs.
• To simplify the discretization of spatial derivatives using a more accessible function space.

Main Methods:

• The Caputo-fractional derivative in time is approximated using a two-step difference method.
• A novel mixed element method is employed for spatial discretization, utilizing the L(2)(Ω)² space instead of H(div; Ω).

Main Results:

• The study establishes a priori error estimates in the L(2)-norm for the scalar unknown u.
• A priori error estimates in the L(2)(Ω)²-norm are derived for the gradient σ.
• Further analysis provides a priori error estimates in the H(1)-norm for the scalar unknown u.

Conclusions:

• The proposed mixed element method offers a computationally efficient and accurate approach for time-fractional PDEs.
• The use of the L(2)(Ω)² space simplifies the method without compromising accuracy.
• The derived error estimates validate the method's effectiveness.