A force-free field (FFF) is determined solely by the normal components of magnetic field and current density on the entire boundary of the domain. Methods employing three components of magnetic field suffer from overspecification of boundary conditions and/or a nonzero divergence-B problem. A vector potential formulation eliminates the latter issue, but introduces difficulties in imposing the normal component of current density at the boundary. This paper proposes four different boundary treatment methods within the vector potential formulation. We conduct a comparative analysis of the vector potential FFF solvers that we have developed incorporating these methods against other FFF codes in different magnetic field representations. Although the vector potential solvers with the new boundary treatments do not outperform our poloidal-toroidal formulation code, they demonstrate comparable or superior performance compared to the optimization code in SolarSoftWare. The methods developed here are expected to be readily applied not only to force-free field computations but also to time-dependent data-driven simulations.

Introduction
Why to Prescribe Jz Instead of B at the Bottom Boundary in the Vector Potential Formulation 
Implementation of Bz and Jz within the Vector Potential Formulation 
    Methods of Adjusting Az and Fixing Axy  
    Methods of Adjusting Axy and Fixing Az  
Numerical Algorithm and the Initial and Boundary Conditions
Force-Free Fields Generated with New Boundary Treatment Methods
    Model TD1 Characterized by a Single Flux Rope  
    Model TD2 Featuring a Bald Patch  
    Model TD3 Featuring Quasi-Separatrix Layers  
Summary and Discussion
Acknowledgments
References

• Sibaek Yi(Department of Astronomy & Space Science, Kyung Hee University, Yongin, Gyeonggi-do 17104, Republic of Korea)
• G. S. Choe(Department of Astronomy & Space Science, Kyung Hee University, Yongin, Gyeonggi-do 17104, Republic of Korea, School of Space Research, Kyung Hee University, Yongin, Gyeonggi-do 17104, Republic of Korea, Institute of Natural Sciences, Kyung Hee University, Yongin, Gyeonggi-do 17104, Republic of Korea) Corresponding author
• Minseon Lee(School of Space Research, Kyung Hee University, Yongin, Gyeonggi-do 17104, Republic of Korea)
• Sunjung Kim(Department of Astronomy & Space Science, Kyung Hee University, Yongin, Gyeonggi-do 17104, Republic of Korea)
• Yeon-Han Kim(Korea Astronomy and Space Science Institute, Daejeon 34055, Republic of Korea)