Uniform Stability and Optimal Error Estimates of a Difference Finite Element Solution for Three-Dimensional Steady MHD Equations

In this paper, a difference finite element (DFE) method is presented for the 3D steady MHD equations. This method approximates the solution pair \(((u,B),p)\) of the 3D steady MHD equations in the \((x,y,z)\)-directions by the DFE solution \[ ((u_h^n,B_h^n),p_h^n)=(∑ₖ₌₀l_3(u_hⁿᵏ,B_hⁿᵏ)φ_k(z),∑ₖ₌₁l_3p_hⁿᵏψ_k(z)) \] of the 3D steady Oseen iterative equations (the 3D steady linearized MHD equations), where \(((u_hⁿᵏ,B_hⁿᵏ),p_hⁿᵏ)\), based on the finite element space pair \(((P_1^b,P_1^b,P_1),(P_1,P_1,P_1))× P_1\), is a series of DFE solution pairs for the 2D steady Oseen iterative equations in the \((x,y)\)-directions. Moreover, we prove the existence and uniqueness of the DFE solution \(((u_h^n,B_h^n),p_h^n)\) under the quasi-uniform mesh condition and deduce first-order convergence with respect to \((σⁿ⁺¹,τ,h)\) of the DFE solution \(((u_h^n,B_h^n),p_h^n)\) to the exact solution \(((u,B),p)\). Finally, numerical experiments illustrate the theoretical results and demonstrate the effectiveness of the proposed method.