The paper presents a new algorithm for computing stable discrete minimal surfaces in $\mathbf{R}^3$, $\mathbf{S}^3$, and $\mathbf{H}^3$, bounded by fixed or free boundary curves. The algorithm does not restrict the genus and can handle singular triangulations. Additionally, an algorithm is introduced to compute a conjugate harmonic map from a given discrete harmonic map, which can be applied to the identity map on a minimal surface to produce its conjugate minimal surface. This process respects the symmetry properties of the boundary curves. The introduction reviews Plateau's Problem, the theoretical existence proofs by Douglas and Radó, and Courant's reformulation. It also discusses numerical methods for computing minimal surfaces, including mean curvature flow and the Dirichlet energy minimization approach. The authors combine aspects of both methods in their algorithm, which involves minimizing the Dirichlet energy step-by-step, avoiding the need for a two-dimensional parameter domain. The paper defines discrete surfaces and minimal surfaces, and provides a detailed explanation of the Dirichlet energy and its minimization. It introduces the concept of discrete harmonic maps and their properties, such as the convex hull property. The minimization algorithm is described, along with its convergence properties and the handling of degenerate triangles. The conjugation algorithm is presented, which constructs the conjugate minimal surface by integrating the dual one-form $*df$ along specific paths. The algorithm ensures that straight lines in the original surface are mapped to planar symmetry curves in the conjugate surface, and vice versa. The paper also discusses topology changes during the minimization process and provides examples illustrating the effectiveness of the algorithms.The paper presents a new algorithm for computing stable discrete minimal surfaces in $\mathbf{R}^3$, $\mathbf{S}^3$, and $\mathbf{H}^3$, bounded by fixed or free boundary curves. The algorithm does not restrict the genus and can handle singular triangulations. Additionally, an algorithm is introduced to compute a conjugate harmonic map from a given discrete harmonic map, which can be applied to the identity map on a minimal surface to produce its conjugate minimal surface. This process respects the symmetry properties of the boundary curves. The introduction reviews Plateau's Problem, the theoretical existence proofs by Douglas and Radó, and Courant's reformulation. It also discusses numerical methods for computing minimal surfaces, including mean curvature flow and the Dirichlet energy minimization approach. The authors combine aspects of both methods in their algorithm, which involves minimizing the Dirichlet energy step-by-step, avoiding the need for a two-dimensional parameter domain. The paper defines discrete surfaces and minimal surfaces, and provides a detailed explanation of the Dirichlet energy and its minimization. It introduces the concept of discrete harmonic maps and their properties, such as the convex hull property. The minimization algorithm is described, along with its convergence properties and the handling of degenerate triangles. The conjugation algorithm is presented, which constructs the conjugate minimal surface by integrating the dual one-form $*df$ along specific paths. The algorithm ensures that straight lines in the original surface are mapped to planar symmetry curves in the conjugate surface, and vice versa. The paper also discusses topology changes during the minimization process and provides examples illustrating the effectiveness of the algorithms.