Investigation of Third-Order Refinement Iterative Methods for Solving Linear Systems in Engineering Applications

The numerical solution of linear systems is fundamental in engineering fields, such as structural analysis, heat transfer, fluid mechanics, and electrical circuit modeling. These applications typically generate systems of the form Tx = u, which may be large, sparse, and moderately ill-conditioned. Refinement-based iterative techniques have been developed to improve convergence characteristics of stationary schemes. This study focuses on the application of existing third refinement iterative methods to engineering linear system problems, with particular emphasis on evaluating their relative computational performance. Engineering-based linear systems were formulated from representative application models and expressed in standard matrix form. Selected third refinement methods, specifically the Third Refinement Jacobi (TRJ) and Third Refinement Gauss-Seidel (TRGS) schemes, were implemented and applied to the test problems. Convergence analysis was conducted using spectral radius criteria derived from the associated iteration matrices. Performance assessment was based on spectral radius, iteration number required to meet a prescribed tolerance, CPU time, and convergence rate. Results were systematically organized in comparative tables. Numerical finding indicates clear performance variation between the refinement schemes. Methods with smaller spectral radii required fewer iterations and reduced computational time. In particular, TRGS demonstrated faster convergence and improved computational efficiency compared to TRJ for the tested engineering problem. The study confirms that third refinement iterative methods are effective for solving engineering linear systems, with performance strongly influenced by spectral properties. These methods provide reliable and computationally efficient solvers for practical engineering applications.