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Fractional spectral collocation (FSC) method based on fractional Lagrange interpolation has recently been proposed to solve fractional differential equations. Numerical experiments show that the linear systems in FSC become extremely…
The Allen--Cahn equation is one of fundamental equations of phase-field models, while the logarithmic Flory--Huggins potential is one of the most useful energy potentials in various phase-field models. In this paper, we consider numerical…
In this thesis, the numerical solution of three different classes of problems have been studied. Specifically, new techniques have been proposed and their theoretical analysis has been performed, accompanied by a wide set of numerical…
We propose a fast and stable method for constructing matrix approximations to fractional integral operators applied to series in the Chebyshev fractional polynomials. This method utilizes a recurrence relation satisfied by the fractional…
In this paper, we carry out stability and error analyses for two first-order, semi-discrete time stepping schemes, which are based on the newly developed Invariant Energy Quadratization approach, for solving the well-known Cahn-Hilliard and…
This paper is concerned with the design, analysis and implementation of preconditioning concepts for spectral Discontinuous Galerkin discretizations of elliptic boundary value problems. While presently known techniques realize a growth of…
In this paper, we analyze the spectra of the preconditioned matrices arising from discretized multi-dimensional Riesz spatial fractional diffusion equations. The finite difference method is employed to approximate the multi-dimensional…
We explore a scaled spectral preconditioner for the efficient solution of sequences of symmetric and positive-definite linear systems. We design the scaled preconditioner not only as an approximation of the inverse of the linear system but…
A finite element scheme for an entirely fractional Allen-Cahn equation with non-smooth initial data is introduced and analyzed. In the proposed nonlocal model, the Caputo fractional in-time derivative and the fractional Laplacian replace…
This paper discusses the spectral collocation method for numerically solving nonlocal problems: one dimensional space fractional advection-diffusion equation; and two dimensional linear/nonlinear space fractional advection-diffusion…
A finite difference numerical scheme is proposed and analyzed for the Cahn-Hilliard-Stokes system with Flory-Huggins energy functional. A convex splitting is applied to the chemical potential, which in turns leads to the implicit treatment…
In this work, initial-boundary value problems for the time-fractional Airy equation are considered on different intervals. We study the properties of potentials for this equation and, using these properties, construct solutions to the…
This paper studies the spectral properties of large matrices and the preconditioning of linear systems, arising from the finite difference discretization of a time-dependent space-fractional diffusion equation with a variable coefficient…
We propose and analyse new stabilized time marching schemes for Phase Fields model such as Allen-Cahn and Cahn-Hillard equations, when discretized in space with high order finite differences compact schemes. The stabilization applies to…
In this paper, we consider the numerical approximations for the fourth order Cahn-Hilliard equation with concentration dependent mobility, and the logarithmic Flory-Huggins potential. One challenge in solving such a diffusive system…
A time-fractional Allen-Cahn equation with volume constraint is first proposed by introducing a nonlocal time-dependent Lagrange multiplier. Adaptive linear second-order energy stable schemes are developed for the proposed model by…
We develop a fast solver for the spectral element method (SEM) applied to the two-sided fractional diffusion equation on uniform, geometric and graded meshes. By approximating the singular kernel with a degenerate kernel, we construct a…
A two-step preconditioned iterative method based on the Hermitian/Skew-Hermitian splitting is applied to the solution of nonsymmetric linear systems arising from the Finite Element approximation of convection-diffusion equations. The…
This paper introduces the sparsifying preconditioner for the pseudospectral approximation of highly indefinite systems on periodic structures, which include the frequency-domain response problems of the Helmholtz equation and the…
The aim of this contribution is to address the convergence study of a time and space approximation scheme for an Allen-Cahn problem with constraint and perturbed by a multiplicative noise of It\^o type. The problem is set in a bounded…