In this article we are interested in the numerical computation of spectra of non-self adjoint quadratic operators, in two and three spatial dimensions. Indeed, in the multidimensional case very few results are known on the location of the eigenvalues. This leads to solve nonlinear eigenvalue problems. In introduction we begin with a review of theoretical results and numerical results obtained for the one dimensional case. Then we present the numerical methods developed to compute the spectra (finite difference discretization) for the two and three dimensional cases. The numerical results obtained are presented and analyzed. One difficulty here is that we have to compute eigenvalues of strongly non-self-adjoint operators which are unstable. This work is in continuity of a previous work in one spatial dimension [3].

In this work, we modify a conjugate gradient (CG) method recently proposed in the literature, where a PRP conjugate gradient method is modified using trust region. Particularly, we propose a hybrid CG method that incorporates the parameters $β^{PRP},$ $β^{FR}$ and $β^{CD},$ and this new search direction satisfies both the trust region feature and the sufficient descent conditions. Furthermore, under suitable conditions the developed method is proved to be globally convergent. The method is tested on some benchmark problems from the literature and numerical results show that it is quite efficient in solving large scale problems.

We consider a Cahn-Hilliard gradient flow model with a free energy functional, which contains a non-local term in addition to linear and non-linear local terms. The non-local terms can be based on smooth and weakly singular kernel operators. We establish the well-posedness of this problem, construct an unconditional energy stable scheme, and carry out a stability and convergence analysis. Several numerical results are presented to illustrate the efficiency and robustness of the proposed scheme.

In this paper, we develop the method of fundamental solutions (MFS) for solving boundary value problems in the field of optical fluorescence. The governing system of diffusion-absorption equations for the excitation and emission fluences is transformed into a single fourth-order partial differential equation whose fundamental solution can be expressed as the difference of two fundamental solutions of the complex Helmholtz equation. The numerically obtained results confirm the accuracy of the MFS when compared with an available analytical solution. Numerical results are also provided for a physical application in optical fluorescence. Furthermore, extensions to three dimensions along with numerical verification are performed.

The efficiency of three Krylov subspace methods with their ILU0-preconditioned version in solving the systems with the nondiagonal sparse matrix is examined. The systems have arisen from the discretization of Poisson’s equation using the 4th and 6th-order compact schemes. Four matrix-vector multiplication techniques based on four sparse matrix storage schemes are considered in the algorithm of the Krylov subspace methods and their effects are explored. The convergence history, error reduction, iteration-resolution relation and CPU-time are addressed. The efficacy of various methods is evaluated against a benchmark scenario in which the conventional second-order central difference scheme is employed to discretize Poisson’s equation. The Krylov subspace methods, paired with four distinct matrix-vector multiplication strategies across three discretization approaches, are tested and implemented within an incompressible fluid flow solver to solve the elliptic segment of the equations. The resulting solution process CPU-time surface gives a new vision regarding speeding up a CFD code with proper selection of discretization stencil and matrix-vector multiplication technique.