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By applying a simple symmetry reduction on a two-layer liquid model, a nonlocal counterpart of it is obtained. Then a general form of nonlocal nonlinear Schrodinger (NNLS) equation with shifted parity, charge-conjugate and delayed time…
Developing robust simulation tools for problems involving multiple mathematical scales has been a subject of great interest in computational mathematics and engineering. A desirable feature to have in a numerical formulation for multiscale…
This work introduces a methodology to solve ordinary differential equations using the Schur decomposition of the linear representation of the differential equation. This is done by first transforming the system into an upper triangular…
We consider the study of a numerical scheme for an initial- and Dirichlet boundary- value problem for a nonlinear Schr\"odinger equation. We approximate the solution using a, local (non-uniform) two level scheme in time (see C. Besse [6]…
Highly resolved finite element simulations of a laser beam welding process are considered. The thermomechanical behavior of this process is modeled with a set of thermoelasticity equations resulting in a nonlinear, nonsymmetric saddle point…
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The spatial discretization of the magnetic vector potential formulation of magnetoquasistatic field problems results in an infinitely stiff differential-algebraic equation system. It is transformed into a finitely stiff ordinary…
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We establish the existence of infinitely many nonnegative, segregated solutions for the sublinearly coupled Schr\"odinger system \begin{equation*} \left\{\begin{aligned}-\Delta u+K_1(x)u&=\mu u^{p-1}+ (\sigma_1+1)\beta…
Convergence failure and slow convergence rates are among the biggest challenges with solving the system of non-linear equations numerically. Although mitigated, such issues still linger when using strictly small time steps and…
Additive overlapping Schwarz Methods are iterative methods of the domain decomposition type for the solution of partial differential equations. Numerical and parallel scalability of these methods can be achieved by adding coarse levels. A…
This paper presents a mathematical foundation for physical models in nonlinear optics through the lens of evolutionary equations. It focuses on two key concepts: well-posedness and exponential stability of Maxwell equations, with models…
In this paper we prove the existence of infinitely many saddle-shaped positive solutions for non-cooperative nonlinear elliptic systems with bistable nonlinearities in the phase-separation regime. As an example, we prove that the system \[…
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We develop a novel iterative solution method for the incompressible Navier-Stokes equations with boundary conditions coupled with reduced models. The iterative algorithm is designed based on the variational multiscale formulation and the…
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Incompressible fluid flow problems appear frequently in different applications. The discretization of such problems may result in large and ill-conditioned systems of linear equations. We consider the case of the Stokes equations…
We revisit the following nonlinear Schr\"odinger system \begin{align*}\begin{cases} -\epsilon^{2}\Delta u +P(x) u= \mu_1 u^3 +\beta uv^2, &~\text{in}\;\mathbb {R}^3,\\ -\epsilon^{2}\Delta v+Q(x) v= \mu_2 v^3 +\beta u^2v,…