Brouwer Fixed Point Theorem Research Articles

Unconditional superconvergence analysis of a new energy stable nonconforming BDF2 mixed finite element method for BBM–Burgers equation

The focus of this paper is to establish a new energy stable 2-step backward differentiation formula (BDF2) fully-discrete mixed finite element method (MFEM) for the Benjamin–Bona–Mahony–Burgers (BBM–Burgers) equation with the nonconforming rectangular EQ1rot element and zero-order Raviart–Thomas (R–T) element (EQ1rot/Q10×Q01). Based on the energy stable property, the existence and uniqueness of the numerical solution are proved by the Brouwer fixed point theorem. Subsequently, with the assistance of some typical properties of this element pair and the interpolation post-processing approach, the unconditional superclose and superconvergence results with O(h2+τ2) are obtained directly without any constraints between the spatial partition parameter h and the time step τ. It is worthy to mention that the method and analysis presented herein are very different from the time–space splitting approach utilized in the previous studies and simplify the implement. Finally, two numerical experiments are executed to validate the theoretical analysis.

An adaptive hybridizable discontinuous Galerkin method for Darcy–Forchheimer flow in fractured porous media

In this paper, we consider an adaptive hybridizable discontinuous Galerkin (HDG) method based on the discrete fracture model for approximation of a single-phase flow in fractured porous media. We are interested in the case that the flow rate in fracture is large enough to justify the use of Forchheimer’s law for modeling flow within the fracture, while Darcy’s law is applied to the surrounding matrix. The HDG method could be designed to simulate the flow in porous media with reduced fractures which consist of many straight lines or planes. More specifically, we use piecewise polynomials of degree [Formula: see text] to approximate the velocity and pressure in fracture and surrounding porous media. The existence and uniqueness of discrete solutions are proved by the Brouwer fixed point theorem, and an efficient and reliable a posteriori error estimator is obtained with respect to an energy norm. Moreover, the HDG scheme, the existence and uniqueness of discrete solutions, and the a posteriori error estimates are also extended to the problem with non-planar, embedded, and intersecting fractures. Finally, several numerical examples are provided to validate the performance of the obtained a posteriori error estimator.

The mass- and energy-conserving relaxation virtual element method for the nonlinear Schrödinger equation

This paper develops a conservative relaxation virtual element method for the nonlinear Schrödinger equation on polygonal meshes. The advantage of this method is to build the virtual element space where the basis functions do not need to be explicitly defined for each local element, and the bilinear forms and nonlinear terms can be computed by using elementwise polynomial projections and pre-defined degrees of freedom. Furthermore, the constructed schemes ensure the conservation of both mass and energy in discrete senses. By using the Brouwer fixed point theorem, we prove the unique solvability of the fully discrete scheme. Finally, some numerical experiments are implemented to verify the theoretical results.

Structure preserving FEM for the perturbed wave equation of quantum mechanics

The construction and analysis of structure-preserving finite element method (FEM) for computing the perturbed wave equation of quantum mechanics are demonstrated. Firstly, a new fully discrete system is built and proved conservative in the sense of the energy. Meanwhile, the boundedness of the numerical solution is derived. Secondly, the existence and uniqueness of the solution are obtained with the help of the Brouwer fixed-point theorem and some special splitting technique. Thirdly, we provide a comprehensive superclose analysis, offering the global superconvergent result. Finally, numerical results are presented to illustrate the theoretical analysis.

Some Boundary Problems, in Sobolev Spaces with Constant Exponents, Governed by the Nonlinear Operator of Plate Vibrations

In this paper, we aim to investigate certain nonlinear boundary problems within Sobolev spaces, where the exponents remain constant. We focus on the dynamically modified operator, incorporating a viscosity term into the nonlinear vibrations of plates. Vibrating plates have a broad range of applications. To address user requirements comprehensively, we've taken into account factors such as the geometric configuration, material density, plate thickness, and Poisson's ratio. After formulating the problems, our method involves converting them into hyperbolic-type nonlinear problems. In this study, we examine six boundary value problems, establishing existence and uniqueness theorems for each. Lastly, we establish the existence of a solution for the stationary problem by employing a variation of Brouwer's fixed point theorem.

Multimode function multistability of Cohen-Grossberg neural networks with Gaussian activation functions and mixed time delays

<abstract><p>This paper explores multimode function multistability of Cohen-Grossberg neural networks (CGNNs) with Gaussian activation functions and mixed time delays. We start by using the geometrical properties of Gaussian functions. The state space is partitioned into $ 3^\mu $ subspaces, where $ 0\le \mu\le n $. Moreover, through the utilization of Brouwer's fixed point theorem and contraction mapping, some sufficient conditions are acquired to ensure the existence of precisely $ 3^\mu $ equilibria for $ n $-dimensional CGNNs. Meanwhile, there are $ 2^\mu $ and $ 3^\mu-2^\mu $ multimode function stable and unstable equilibrium points, respectively. Ultimately, two illustrative examples are provided to confirm the efficacy of theoretical results.</p></abstract>

An AC-Feasible Linear Model in Distribution Networks With Energy Storage

With the increasing deployment of distributed energy resources (DERs), dispatching DERs subject to operational constraints in distribution networks draws much attention. One challenge is the non-convexities in 1) system-wide AC power flow constraints and 2) the individual complementarity constraint of energy storage. To resolve this challenge, this paper studies an AC-feasible linear model in distribution networks with energy storage, including its formulation, analysis, and applications. First, an AC-feasible linear model is formulated as a set of linear constraints on controllable DERs and uncontrollable power demand by 1) converting the non-convex system-wide constraints into linear constraints based on the Brouwer's fixed-point theorem and the second-order Taylor expansion, and 2) replacing the non-convex individual complementarity constraint of energy storage with one properly designed linear constraint. Furthermore, to analyze the power demand level at which the proposed linear model can provide a solution, this paper proposes an examination-based projection method under the Monte Carlo framework to handle projections of thousands of dimensions from linear constraints over time periods. Finally, the potential applications of our AC-feasible linear model are discussed. Numerical experiments are conducted in the IEEE 33-bus and 136-bus test systems to demonstrate the effectiveness of the proposed methods.

Unconditional superconvergence analysis of an energy-stable L1 scheme for coupled nonlinear time-fractional prey-predator equations with nonconforming finite element

The constrained nonconforming rotated Q1 (CNQ1rot) finite element is applied to develop an energy-stable L1 fully-discrete scheme through the Caputo fractional derivative approximation for the coupled nonlinear time-fractional prey-predator equations, and the unconditional superconvergence behavior is investigated. In which a novel high-accuracy estimate for the CNQ1rot element is given with the help of the Bramble-Hilbert (B-H) lemma, and the energy stability of the numerical solution is confirmed. Both of them are critical for demonstrating the unique solvability of the proposed scheme via the Brouwer fixed point theorem, and deducing the superclose estimation without any limitations between the spatial division size h and the time step τ. In addition, the above high-accuracy estimate can be extended to anisotropic meshes. Then, the unconditional superconvergence result is obtained by introducing the interpolation post-processing operator. Lastly, the theoretical findings are supported by some numerical experiments.

Unconditional superconvergence analysis of an energy-preserving finite element scheme for nonlinear BBM equation

An energy-preserving Crank-Nicolson (CN) fully-discrete scheme is developed with the nonconforming Quasi-Wilson element for the nonlinear Benjamin-Bona-Mahony (BBM) equation. The existence and uniqueness of the numerical solution are demonstrated by the Brouwer fixed point theorem. Then with the help of the special character of this element, that is, the consistency error can reach order O(h2), one order higher than its interpolation error, and the interpolation post-processing technique, the unconditional supercloseness and superconvergence behavior on quadrilateral meshes are derived rigorously without the restriction between mesh size h and time step τ. Finally, some numerical experiments are carried out to confirm the theoretical analysis.

Unconditional superconvergence analysis of an energy conservation scheme with Galerkin FEM for nonlinear Benjamin–Bona–Mahony equation

In this paper, a general energy conservation Crank–Nicolson (CN) fully-discrete finite element method (FEM) scheme is developed for solving the nonlinear Benjamin–Bona–Mahony (BBM) equation, and the unconditional supercloseness and superconvergence behavior are discussed with the nonconforming EQ1rot element. Different from the time–space splitting technique, the boundedness of numerical solution in the broken H1-norm is obtained directly based on the above energy conservation property, and the well-posedness of numerical solution is proved rigorously through the Brouwer fixed point theorem. Then, with the help of the special characters of this element and the interpolation post-processing technique, the unconditional superclose and superconvergence results are deduced strictly without any restriction between mesh size h and time step τ. Further, our analysis is extended to some other popular conforming and nonconforming finite elements. Finally, two numerical experiments are implemented to confirm the theoretical analysis. It is worth mentioning that this paper not only improve the previous results, but also can be regard as a framework for solving the BBM equation and some other PDEs with Galerkin FEMs.

Quasilinear Dirichlet systems with competing operators and convection

In this paper, we consider a quasi-linear Dirichlet system with possible competing (p,q)-Laplacians and convections. Due to the lack of ellipticity, monotonicity, and variational structure, the standard approaches to the existence of weak solutions cannot be adopted. Nevertheless, through an approximation procedure and a corollary of Brouwer's fixed point theorem we show that the problem admits a solution in a suitable sense.

New Banach spaces-based fully-mixed finite element methods for pseudostress-assisted diffusion problems

In this paper we propose and analyze Banach spaces-based fully-mixed approaches yielding new finite element methods for numerically solving the coupled partial differential equations describing the pseudostress-assisted diffusion of a solute into an elastic material. Two mixed formulations employing the diffusive flux as an additional variable are introduced for the diffusion equation, and the concentration gradient is considered as an auxiliary unknown of the second one of them. The resulting coupled systems are rewritten as equivalent fixed point operator equations, so that the respective unique solvabilities are proved by applying the classical Banach theorem along with the Babuška-Brezzi theory. The nonlinear dependency on the elastic variables of the diffusion coefficient and its source term, as well as the nonlinear dependency on the concentration of the elastic source term, suggest, for appropriate continuous and discrete analyses, that the unknowns be sought in suitable Lebesgue spaces. The associated Galerkin schemes are addressed similarly, and the Brouwer theorem yields the existence of discrete solutions. A priori error estimates are derived for both approaches, and rates of convergence for specific finite element subspaces satisfying the required discrete inf-sup conditions, are established in 2D. Finally, several numerical examples illustrating the performance of the two methods and confirming the theoretical findings, are reported.

Unconditional superconvergence analysis of a modified nonconforming energy stable BDF2 FEM for Sobolev equations with Burgers’ type nonlinearity

The aim of this paper is to develop a modified energy stable 2-step backward differentiation formula (BDF2) fully discrete scheme and study its superconvergence behavior with the nonconforming quadrilateral quasi-Wilson element for the Sobolev equations with Burgers’ type nonlinearity. The existence and uniqueness of the numerical solution are proved by Brouwer fixed point theorem. Then, different from the so-called time–space splitting technique, some typical properties of this element especially the consistency error can reach order O(h2), one order higher than its interpolation error, and the interpolation post-processing approach are used to derive the unconditional supercloseness and superconvergence results on quadrilateral meshes without the restriction between the mesh size h and the time step τ. At last, a numerical example is carried out and a comparison with the Wilson element is also given to confirm the theoretical analysis.

Unconditional superconvergence analysis of an energy-stable C–N fully discrete scheme for the nonlinear magnetic diffusion model with memory

The unconditional superconvergence behavior of an energy-stable Crank–Nicolson (C–N) fully discrete scheme with nonconforming EQ1rot finite element method (FEM) is investigated for the nonlinear integro-differential system associated with the diffusion of the magnetic field. The stability of the numerical solution in the broken H1-norm is proved by the Gagliardo–Nirenberg (G–N) inequality, and the unique solvability is demonstrated by the Brouwer fixed point theorem. In which the boundness of the L∞-norm and the inverse inequality are eliminated. The combination of the Green formula and two special properties of EQ1rot element leads to the unconditional superconvergence estimate without any restriction between the time step and space partition size. Finally, the validity of the theoretical findings is supported by some numerical results.

Nonconforming modified Quasi-Wilson finite element method for convection–diffusion-reaction equation

The nonconforming modified Quasi-Wilson finite element method (FEM) is employed to investigate the unconditional superconvergence behavior of the convection–diffusion-reaction equation for the linearized energy stable Backward-Euler (BE) and the second order Backward differentiation formula (BDF2) fully discrete schemes on arbitrary quadrilateral meshes. Then, the energy stabilities of the discrete solutions for the proposed schemes are demonstrated through the mathematical induction, which is crucial to proving the unique solvabilities with Brouwer fixed point theorem. Based on the above achievements and the two typical characteristics of the element: one is that its consistency error in the broken H1-norm can be estimated as order O(h2) when the exact solution belongs to H3(Ω), which is one order higher than its interpolation error, and the other is that the nonconforming part of the function in the finite element space is orthogonal to the bilinear polynomials on each element of the subdivision in the integration sense, the superclose estimates without any restriction between spatial partition parameter h and the time step τ are obtained. Furthermore, the superconvergence properties for the aforementioned schemes are derived by the interpolation post-processing technique. Finally, the theoretical analysis is justified by the provided numerical experiments.

Two‐level stabilized finite volume method for the stationary incompressible magnetohydrodynamic equations

AbstractIn this paper, a two‐level stabilized finite volume method is developed and analyzed for the steady incompressible magnetohydrodynamic (MHD) equations. The linear polynomial space is used to approximate the velocity, pressure and magnetic fields, and two local Gauss integrations are introduced to overcome the restriction of discrete inf‐sup condition. Firstly, the existence and uniqueness of the solution of the discrete problem in the stabilized finite volume method are proved by using the Brouwer's fixed point theorem. ‐stability results of numerical solutions are also presented. Secondly, optimal error estimates of numerical solutions in and ‐norms are established by using the energy method and constructing the corresponding dual problem. Thirdly, the stability and convergence of two‐level stabilized finite volume method for the stationary incompressible MHD equations are provided. Theoretical findings show that the two‐level method has the same accuracy as the one‐level method with the mesh sizes . Finally, some numerical results are provided to identify with the established theoretical findings and show the performances of the considered numerical schemes.

Specific Features of the Angular Distribution of Electromagnetic Radiation of the Kerr Black Hole

By analyzing algebraically special solutions of the Maxwell equations in the Kerr space-time, we obtain the exact expressions for the polarization characteristics of electromagnetic waves emitted from the vicinity of a black hole. We revealed the asymmetry of dependence of the ellipticity angle on the polar angle for the fundamental mode and the first harmonics of polarized radiation. This creates a basis for the new method of evaluation of the intrinsic angular momentum of the Kerr black hole. It is shown that the existence of singular points of the solution in a local orthonormal frame is a consequence of the Poincaré–Brouwer theorem.

Brouwer fixed point theorem in strictly star-shaped sets

Brouwer fixed point theorem in strictly star-shaped sets

On a class of nonlinear time‐fractional pseudo‐parabolic equations with bounded delay

In the present paper, we study a class of nonlinear time‐fractional pseudo‐parabolic equations with bounded time‐delay. Firstly, we analyze the well‐posedness of the problem by the well‐known Galerkin's method and some compactness criteria. Next, the existence of the stationary solutions is established by utilizing a lemma of ours, which is a corollary of the Brouwer fixed point theorem. Finally, with regard to the stability analysis of the stationary solutions, we prove the local stability via the direct method of energy estimations and show the asymptotic stability by the construction of appropriate Lyapunov functionals.

A stabilized finite element method for the Stokes–Temperature coupled problem

In this work, we introduce and analyze a new stabilized finite element scheme for the Stokes–Temperature coupled problem. This new scheme allows equal order of interpolation to approximate the quantities of interest, i.e. velocity, pressure, temperature, and stress. We analyze an equivalent variational formulation of the coupled problem inspired by the ideas proposed in [3]. The existence of the discrete solution is proved, decoupling the proposed stabilized scheme and using the help of continuous dependence results and Brouwer's theorem under the standard assumption of sufficiently small data. Optimal convergence is proved under classic regularity assumptions of the solution. Finally, we present some numerical examples to show the quality of our scheme, in particular, we compare our results with those coming from a standard reference in geosciences described in [38].