Related papers: Solving nonlinear Klein-Gordon equations on unboun…
The mass of a scalar field mediating a fifth force is tightly constrained by experiments. We show, however, that adding a quartic self-interaction for such a scalar makes most tests much less constraining: the non-linear equation of motion…
In this paper we analyze a class of trace finite element methods (TraceFEM) for the discretization of vector-Laplace equations. A key issue in the finite element discretization of such problems is the treatment of the constraint that the…
We study numerical methods for the rotating nonlinear Klein-Gordon (RKG) equation, a fundamental model in relativistic quantum physics, which exhibits highly oscillatory multiscale behavior due to the presence of a small parameter…
In this work, approximate solutions to the nonlinear Klein-Gordon equation are constructed by means of the Galerkin method. Specifically, it is shown how the dynamics of a real scalar field in $1+1$ dimensions subjected to Dirichlet…
The finite element method is used to approximately solve boundary value problems for differential equations. The method discretises the parameter space and finds an approximate solution by solving a large system of linear equations. Here we…
In this paper, we study the existence, regularity, and approximation of the solution for a class of nonlinear fractional differential equations. {In order to do this}, suitable variational formulations are defined for a nonlinear boundary…
Chameleon scalar fields can screen their associated fifth forces from detection by changing their mass with the local density. These models are an archetypal example of a screening mechanism, and have become an important target for both…
A nonlinear Helmholtz equation (NLH) with high wave number and Sommerfeld radiation condition is approximated by the perfectly matched layer (PML) technique and then discretized by the linear finite element method (FEM).…
Finite size scaling for the Schr\"{o}dinger equation is a systematic approach to calculate the quantum critical parameters for a given Hamiltonian. This approach has been shown to give very accurate results for critical parameters by using…
The paper studies an Allen-Cahn-type equation defined on a time-dependent surface as a model of phase separation with order-disorder transition in a thin material layer. By a formal inner-outer expansion, it is shown that the limiting…
The Fast Multipole Method (FMM) provides a highly efficient computational tool for solving constant coefficient partial differential equations (e.g. the Poisson equation) on infinite domains. The solution to such an equation is given as the…
We formulate, analyse, and implement a discontinuous Galerkin finite element method (DG-FEM) for the approximation of the solution of an elliptic boundary value problem in a domain with fractal boundary. We consider the case of the Poisson…
The finite element method(FEM) is applied to bound leading eigenvalues of Laplace operator over polygonal domain. Compared with classical numerical methods, most of which can only give concrete eigenvalue bounds over special domain of…
The chameleon model is a modified gravity theory that introduces an additional scalar field that couples to matter through a conformal coupling. This `chameleon field' possesses a screening mechanism through a nonlinear self-interaction…
Light scalar fields are expected to arise in theories of high energy physics (such as string theory), and find phenomenological motivations in dark energy, dark matter, or neutrino physics. However, the coupling of light scalar fields to…
In this paper we use the GeneralizedMultiscale Finite ElementMethod (GMsFEM) framework, introduced in [20], in order to solve nonlinear elliptic equations with high-contrast coefficients. The proposed solution method involves linearizing…
Nonlocal models allow for the description of phenomena which cannot be captured by classical partial differential equations. The availability of efficient solvers is one of the main concerns for the use of nonlocal models in real world…
Multiscale Finite Element Methods (MsFEM) are finite element type approaches dedicated to multiscale problems. They first compute local, oscillatory, problem-dependent basis functions which generate a specific discretization space, and next…
The $f(R)$ gravity can be cast into the form of a scalar-tensor theory, and scalar degree of freedom can be suppressed in high-density regions by the chameleon mechanism. In this article, for the general $f(R)$ gravity, using a…
Nonexistence of global weak solutions of Klein-Gordon equations with gauge variant semilinear terms are considered in Friedmann-Lema\^itre-Robertson-Walker spacetimes. Effects of spatial expansion or contraction on the solutions are studied…