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The computational analysis of fiber network fracture is an emerging field with application to paper, rubber-like materials, hydrogels, soft biological tissue, and composites. Fiber networks are often described as probabilistic structures of interacting one-dimensional elements, such as truss-bars and beams. Failure may then be modeled as strong discontinuities in the displacement field that are directly embedded within the structural finite elements. As for other strain-softening materials, the tangent stiffness matrix can be non-positive definite, which diminishes the robustness of the solution of the coupled (monolithic) two-field problem. Its uncoupling, and thus the use of a staggered solution method where the field variables are solved alternatingly, avoids such difficulties and results in a stable, but sub-optimally converging solution method. In the present work, we evaluate the staggered against the monolithic solution approach and assess their computational performance in the analysis of fiber network failure. We then propose a hybrid solution technique that optimizes the performance and robustness of the computational analysis. It represents a matrix regularization technique that retains a positive definite element stiffness matrix while approaching the tangent stiffness matrix of the monolithic problem. The approach is general and may also accelerate the computational analysis of other failure problems.