- for systems with non-symmetrical and Gradient non-singular matrices 7.
For the numerical solution of the velocity and stress linear systems of equations, we used the biconjugate
gradient stabilized method; for the pressure, the conjugate gradient method was used together with a Generalized Geometric-Algebraic Multi-Grid (GAMG) preconditioner.
gradient stabilized method (BiCGSTAB) is adopted as the iterative solver for the matrix equations, and the threshold of the iterative stopping criterion for residuum is set to 0.001.
The preconditioner of diagonal incomplete LU and the biconjugate
gradient method (PBICG) by Jacobs  is used for solving momentum equations with a local accuracy of [10.sup.-8] at each step.
Once all the terms for all the finite volumes within the domain have been assembled into [A.sup.k+1] and [g.sup.k+1], the equation [A.sup.k+1][[??].sup.k+1] = [g.sup.k+1] is solved using a biconjugate
gradient stabilized method to give the updated fluid concentration, [[??].sup.k+1] , at each discrete spatial point in [[??].sub.f], while the updated adsorbed concentration, [[??].sup.k+1], is given by (3.4).
However, there are a number of variants to the basic Conjugate Gradient algorithm (Conjugate Residuals, Biconjugate
Gradient, Stabilized Biconjugate
Gradient, Lanczos method, Generalized Minimal Residuals (GMINRES), Bi-Lanczos, Conjugate Gradient Squared, Quasi Minimal Residuals, etc.; most of them belong to a family of algorithms called Krylov subspace algorithms [107, 108]), none of which have been tested in EM.
The embedding theorem of a locally convex space into its biconjugate
space has played a crucial role in the study of semireflexivity and reflexivity of a locally convex space.
This study carefully explores the application of the Generalized Minimal Residual (GMRES) and the BiConjugate
Gradient (BiCGstab) methods to these problems, and recommends using the latter method with an incomplete LU preconditioning.
Mittra, "The biconjugate
gradient method for electromagnetic scattering," IEEE Trans.
Liu, "Three-dimensional electromagnetic nonlinear inversion in layered media by a hybrid diagonal tensor approximation: stabilized biconjugate
gradient fast Fourier transform method," Waves in Random and Complex Media, vol.
Linear algebra exploits acceleration with the use ofDiagonal Incomplete LU factorization to precondition BiConjugate
The following methods can be used to solve a heat transfer-related linear equation system: generalized minimum residual iterative method, flexible generalized minimum residual iterative method, biconjugate
gradient stabilized iterative method, conjugate gradients iterative method .