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In this article we study a coupled system of differential equations with Allen-Cahn type non-linearity. Motivated by physical phenomena one of the unknowns in the system is accompanied by a singular perturbation parameter ${\epsilon}^2$ .…
We establish a framework which allows one to construct novel schemes for measurement-based quantum computation. The technique further develops tools from many-body physics - based on finitely correlated or projected entangled pair states -…
We prove various results in infinite-dimensional differential calculus which relate differentiability properties of functions and associated operator-valued functions (e.g., differentials). The results are applied in two areas: 1. in the…
In this paper we consider a fully third order nonlinear boundary value problem which is of great interest of many researchers. First we establish the existence, uniqueness of solution. Next, we propose simple iterative methods on both…
We introduce a variational time discretization for the multi-dimensional gas dynamics equations, in the spirit of minimizing movements for curves of maximal slope. Each timestep requires the minimization of a functional measuring the…
We consider the variational complex on infinite jet space and the complex of variational derivatives for Lagrangians of multidimensional paths and study relations between them. The discussion of the variational (bi)complex is set up in…
We propose a collocation method based on multivariate polynomial splines over triangulation or tetrahedralization for the numerical solution of partial differential equations. We start with a detailed explanation of the method for the…
We introduce vertex block descent, a block coordinate descent solution for the variational form of implicit Euler through vertex-level Gauss-Seidel iterations. It operates with local vertex position updates that achieve reductions in global…
We study a numerical approximation for a nonlinear variable-order fractional differential equation via an integral equation method. Due to the lack of the monotonicity of the discretization coefficients of the variable-order fractional…
A number of important results of studying large deformations of hyper-elastic shells are obtained using discrete methods of mathematical physics. In the present paper, using the variational method for solving nonlinear boundary problems of…
We study numerical methods for the nonlinear partial differential equation that governs the motion of level sets by affine curvature. We show that standard finite difference schemes are nonlinearly unstable. We build convergent finite…
An abstract 2nd-order evolution equation or inclusion is discretised in time in such a way that the energy is conserved at least in qualified cases, typically in the cases when the governing energy is component-wise quadratic or…
We introduce a sum-of-squares SDP hierarchy approximating the ground-state energy from below for quantum many-body problems, with a natural quantum embedding interpretation. We establish the connections between our approach and other…
A brief review is presented of the scaling of complex fluids, polymers and polyelectrolytes in solution and in confined geometry, in thermodynamical, structural and rheology properties using equilibrium and nonequilibrium dissipative…
A nonlinear block-coupled Finite Volume methodology is developed for large displacement and large strain regime. The new methodology uses the same normal and tangential face derivative discretisations found in the original fully coupled…
The paper presents a model of lateral phase separation in a two component material surface. The resulting fourth order nonlinear PDE can be seen as a Cahn-Hilliard equation posed on a time-dependent surface. Only elementary tangential…
New families of fourth-order composition methods for the numerical integration of initial value problems defined by ordinary differential equations are proposed. They are designed when the problem can be separated into three parts in such a…
Biharmonic wave equations are of importance to various applications including thin plate analyses. In this work, the numerical approximation of their solutions by a $C^1$-conforming in space and time finite element approach is proposed and…
This paper promotes the differential method as a new fruitful strategy for estimating a ground-state energy of a many-body system. The case of an arbitrary number of attractive Coulombian particles is specifically studied and we make some…
A summary of number structure scaling is followed by a description of the effects of number scaling in nonrelativistic quantum mechanics. The description extends earlier work to include the effects on the states of two or more interacting…