Case Of Variable Coefficients Research Articles

Analysis of nonconforming IFE methods and a new scheme for elliptic interface problems

In this paper, an important discovery has been found for nonconforming immersed finite element (IFE) methods using the integral values on edges as degrees of freedom for solving elliptic interface problems. We show that those IFE methods without penalties are not guaranteed to converge optimally if the tangential derivative of the exact solution and the jump of the coefficient are not zero on the interface. A nontrivial counter example is also provided to support our theoretical analysis. To recover the optimal convergence rates, we develop a new nonconforming IFE method with additional terms locally on interface edges. The new method is parameter-free which removes the limitation of the conventional partially penalized IFE method. We show the IFE basis functions are unisolvent on arbitrary triangles which is not considered in the literature. Furthermore, different from multipoint Taylor expansions, we derive the optimal approximation capabilities of both the Crouzeix–Raviart and the rotated-Q1 IFE spaces via a unified approach which can handle the case of variable coefficients easily. Finally, optimal error estimates in both H1- and L2-norms are proved and confirmed with numerical experiments.

Method of Particular Solutions for Second-Order Differential Equation with Variable Coefficients via Orthogonal Polynomials

In this paper, with classic Legendre polynomials, a method of particular solutions (MPS, for short) is proposed to solve a kind of second-order differential equations with a variable coefficient on a unit interval. The particular solutions, satisfying the natural Dirichlet boundary conditions, are constructed with orthogonal Legendre polynomials for the variable coefficient case. Meanwhile, we investigate the a-priori error estimates of the MPS approximations. Two a-priori error estimations in H 1 - and L ∞ -norms are shown to depict the convergence order of numerical approximations, respectively. Some numerical examples and convergence rates are provided to validate the merits of our proposed meshless method.

Increasing stability for the inverse source problem in elastic waves with attenuation

AbstractThis paper is concerned with the increasing stability of the inverse source problem for the elastic wave equation with attenuation in three dimensions. The stability estimate consists of the Lipschitz type data discrepancy and the high frequency tail of the source function, where the latter decreases as the upper bound of the frequency increases. The stability also shows exponential dependence on the attenuation coefficient. The ingredients of the analysis include Carleman estimates and time decay estimates for the elastic wave equation to obtain an exact observability bound, and the study of the resonance-free region and an upper bound of the resolvent in this region for the elliptic operator with respect to the complex frequency. The advantage of the method developed in this work is that it can be used to study the case of variable attenuation coefficient.

Numerical solution of kinetic SPDEs via stochastic Magnus expansion

In this paper, we show how the Itô-stochastic Magnus expansion can be used to efficiently solve stochastic partial differential equations (SPDE) with two space variables numerically. To this end, we will first discretize the SPDE in space only by utilizing finite difference methods and vectorize the resulting equation exploiting its sparsity.As a benchmark, we will apply it to the case of the stochastic Langevin equation with constant coefficients, where an explicit solution is available, and compare the Magnus scheme with the Euler–Maruyama scheme. We will see that the Magnus expansion is superior in terms of both accuracy and especially computational time by using a single GPU and verify it in a variable coefficient case. Notably, we will see speed-ups of order ranging form 20 to 200 compared to the Euler–Maruyama scheme, depending on the accuracy target and the spatial resolution.

Fractional-power approach for the study of elliptic second-order boundary-value problems with variable-operator coefficients in an unbounded domain

In this article, we give new results on the study of elliptic complete abstract second order differential equations with variable operator coefficients under Dirichlet boundary conditions, and set in \(\mathbb{R}_+\). In the framework of Holderian spaces and under some compatibility conditions, we prove the main results on the existence, uniqueness and maximal regularity of the classical solution of this kind of problems which have not been studied in variable coefficients case. We use semigroups theory, fractional powers of linear operators, Dunford's functional calculus and interpolation theory. In this work, we consider some differentiability assumptions on the resolvents of square roots of linear operators.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/89/abstr.html

Analysis of a high-order compact finite difference method for Robin problems of time-fractional sub-diffusion equations with variable coefficients

This paper is concerned with the construction and analysis of a high-order compact finite difference method for a class of time-fractional sub-diffusion equations under the Robin boundary condition. The diffusion coefficient of the equation may be spatially variable and the time-fractional derivative is in the Caputo sense with the order α∈(0,1). A (3−α)th-order numerical formula (called the L2 formula here) without any sub-stepping scheme for the approximation at the first-time level is applied to the discretization of the Caputo time-fractional derivative. A new fourth-order compact finite difference operator is constructed to approximate the variable coefficient spatial differential operator under the Robin boundary condition. By developing a technique of discrete energy analysis, the unconditional stability of the proposed method and its convergence of (3−α)th-order in time and fourth-order in space are rigorously proved for the general case of variable coefficient and for all α∈(0,1). Further approximations are considered for enlarging the applicability of the method while preserving its high-order accuracy. Numerical results are provided to demonstrate the theoretical analysis results.

Spectral analysis of Pk Finite Element matrices in the case of Friedrichs–Keller triangulations via Generalized Locally Toeplitz technology

SummaryIn the present article, we consider a class of elliptic partial differential equations with Dirichlet boundary conditions and where the operator is div(−a(x)∇·), with a continuous and positive over , Ω being an open and bounded subset of , d≥1. For the numerical approximation, we consider the classical Finite Elements, in the case of Friedrichs–Keller triangulations, leading, as usual, to sequences of matrices of increasing size. The new results concern the spectral analysis of the resulting matrix‐sequences in the direction of the global distribution in the Weyl sense, with a concise overview on localization, clustering, extremal eigenvalues, and asymptotic conditioning. We study in detail the case of constant coefficients on Ω=(0,1)2 and we give a brief account in the more involved case of variable coefficients and more general domains. Tools are drawn from the Toeplitz technology and from the rather new theory of Generalized Locally Toeplitz sequences. Numerical results are shown for a practical evidence of the theoretical findings.

Property of oscillation of first order impulsive neutral differential equations with positive and negative coefficients

In this paper, necessary and sufficient conditions for oscillation are obtained, so that every solution of the linear impulsive neutral differential equation with variable delays and variable coefficients oscillates. Most of authors who study the oscillatory criteria of impulsive neutral differential equations, investigate the case of constant delays and variable coefficients. However the points of impulsive in this paper are more general. An illustrate example is given to demonstrate our claim and explain the results.

Analysis of elastic symbols with the Cauchy integral and construction of asymptotic solutions

This paper deals with the elastic wave equation $(D_t^2 - L(x, D_{x'}, D_{x_n})) u(t, x', x_n)=0$ in the half-space $x_n>0$. In the constant coefficient case, it is known that the solution is represented by using the Cauchy integral $\int_c e^{ix_n\zeta} (I-L(\xi', \zeta))^{-1} d\zeta$. In this paper this representation is extended to the variable coefficient case, and an asymptotic solution with the similar Cauchy integral is constructed. In this case, the terms $\partial_x^\alpha \int_c e^{ix_n\zeta} (I-L(x,\xi',\zeta))^{-1} d\zeta$ appear in the inductive process. These do not become lower terms necessarily, and therefore the principal part of asymptotic solution is a little different from the form in the constant coefficient case.

A high-order L2-compact difference method for Caputo-type time-fractional sub-diffusion equations with variable coefficients

A high-order compact finite difference method is proposed for solving a class of time-fractional sub-diffusion equations. The diffusion coefficient of the equation may be spatially variable and the time-fractional derivative is in the Caputo sense with the order α ∈ (0, 1). The Caputo time-fractional derivative is discretized by a (3−α) th-order numerical formula (called the L2 formula here) which is constructed by piecewise quadratic interpolating polynomials but does not require any sub-stepping scheme for the approximation at the first-time level. The variable coefficient spatial differential operator is approximated by a fourth-order compact finite difference operator. By developing a technique of discrete energy analysis, a full theoretical analysis of the stability and convergence of the method is carried out for the general case of variable coefficient and for all α ∈ (0, 1). The optimal error estimate is obtained in the L2 norm and shows that the proposed method has the temporal (3−α) th-order accuracy and the spatial fourth-order accuracy. Further approximations are also considered for enlarging the applicability of the method while preserving its high-order accuracy. Applications are given to three model problems, and numerical results are presented to demonstrate the theoretical analysis results.

Convergence Analysis of Krylov Subspace Spectral Methods for Reaction–Diffusion Equations

Krylov subspace spectral (KSS) methods are explicit time-stepping methods for partial differential equations that are designed to extend the advantages of Fourier spectral methods, when applied to constant-coefficient problems, to the variable-coefficient case. This paper presents a convergence analysis of a first-order KSS method applied to a system of coupled equations for modeling first-order photobleaching kinetics. The analysis confirms what has been observed in numerical experiments—that the method is unconditionally stable and achieves spectral accuracy in space. Further analysis shows that this unconditional stability is not limited to the case in which the leading coefficient is constant.

High-Order Compact Difference Methods for Caputo-Type Variable Coefficient Fractional Sub-diffusion Equations in Conservative Form

A set of high-order compact finite difference methods is proposed for solving a class of Caputo-type fractional sub-diffusion equations in conservative form. The diffusion coefficient of the equation may be spatially variable, and the proposed methods have the global convergence order $$\mathcal{O}(\tau ^{r}+h^{4})$$ , where $$r\ge 2$$ is a positive integer and $$\tau $$ and h are the temporal and spatial steps. Such new high-order compact difference methods greatly improve the known methods in the literature. The local truncation error and the solvability of the methods are discussed in detail. By applying a discrete energy technique to the matrix form of the methods, a rigorous theoretical analysis of the stability and convergence of the methods is carried out for the case of $$2\le r\le 6$$ , and the optimal error estimates in the weighted $$H^{1}$$ , $$L^{2}$$ and $$L^{\infty }$$ norms are obtained for the general case of variable coefficient. Applications are given to two model problems, and some numerical results are presented to illustrate the various convergence orders of the methods.

Fast evaluation and high accuracy finite element approximation for the time fractional subdiffusion equation

In this article, an efficient algorithm for the evaluation of the Caputo fractional derivative and the superconvergence property of fully discrete finite element approximation for the time fractional subdiffusion equation are considered. First, the space semidiscrete finite element approximation scheme for the constant coefficient problem is derived and supercloseness result is proved. The time discretization is based on the L1‐type formula, whereas the space discretization is done using, the fully discrete scheme is developed. Under some regularity assumptions, the superconvergence estimate is proposed and analyzed. Then, extension to the case of variable coefficients is also discussed. To reduce the computational cost, the fast evaluation scheme of the Caputo fractional derivative to solve the fractional diffusion equations is designed. Finally, numerical experiments are presented to support the theoretical results.

Energy decay for the wave equation of variable coefficients with acoustic boundary conditions in domains with nonlocally reacting boundary

In this paper, we consider the wave equation of variable coefficients with acoustic boundary conditions in domains with nonlocally reacting boundary. This work is devoted to prove the energy decay for the wave equation of variable coefficients having some boundary reacting conditions. This paper is to improve our previous result in Ha (2016) by applying the variable coefficients case.

Subsidizing healthcare in the presence of market distortions

The paper develops a three-sector full-employment general equilibrium model for a small open developing economy with exogenous labour market imperfection and a non-traded sector providing healthcare services, the consumption of which generates positive externalities. Our main objective is to show that the optimal consumption subsidy to healthcare, if solely judged from the standpoint of economic growth, is strictly positive (zero) when the production technology of the healthcare sector is of the variable (fixed) coefficient type. However, in the variable coefficient case, the optimal per capita expenditure on healthcare crucially hinges on the degree of labour market imperfection and the quality of services provided by the healthcare sector. The latter result can possibly be considered as a theoretical justification why the magnitude of per capita public spending on healthcare services is significantly lower in the developing countries compared to that in the developed nations. Besides, using the Sen's (1974) index of social welfare that takes into consideration both the growth and income inequality aspects, we have proved that the optimal health subsidy is positive irrespective of the nature of production technology of the healthcare sector. Furthermore, most of these results are found to be valid even in the presence of Harris-Todaro type unemployment. Finally, the results lead to a few important policy implications in the context of the developing countries.

Modeling of cleaning of dust emission’ in fluidized bed building aspiration’ collector

This article describes one of the modern way to reduce dust emissions of pollutions exhausting into the atmosphere at expanded clay aggregates and other similar building materials manufactures applying filtering fluidized granular particulate material bed’ separator with low degree of dust leakage out from one. There is presented quasi-diffusion model featuring of process of cleaning of industrial emissions of dust in devices of tray type with the fluidized and weighted bed. There considered case of variable coefficient of longitudinal hashing intermixing within trough tray type separator in this article. It was made attempt to get meanings value of leakage’ degree dust out from separator. It was obtain in an implicit form. It was obtained and announced some results of the carried-out analysis are intended to get high efficiency of dust removal set up installations to clean emissions of aspiration scheme of the air environmental protection in production of bulk dispersed materials building construction industry.

Virtual Element Method for general second-order elliptic problems on polygonal meshes

We consider the discretization of a boundary value problem for a general linear second-order elliptic operator with smooth coefficients using the Virtual Element approach. As in [A. H. Schatz, An observation concerning Ritz–Galerkin methods with indefinite bilinear forms, Math. Comput. 28 (1974) 959–962] the problem is supposed to have a unique solution, but the associated bilinear form is not supposed to be coercive. Contrary to what was previously done for Virtual Element Methods (as for instance in [L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013) 199–214]), we use here, in a systematic way, the [Formula: see text]-projection operators as designed in [B. Ahmad, A. Alsaedi, F. Brezzi, L. D. Marini and A. Russo, Equivalent projectors for virtual element methods, Comput. Math. Appl. 66 (2013) 376–391]. In particular, the present method does not reduce to the original Virtual Element Method of [L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013) 199–214] for simpler problems as the classical Laplace operator (apart from the lowest-order cases). Numerical experiments show the accuracy and the robustness of the method, and they show as well that a simple-minded extension of the method in [L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013) 199–214] to the case of variable coefficients produces, in general, sub-optimal results.

Low-Rank Solvers for Fractional Differential Equations

Many problems in science and technology can be cast using differential equations with both fractional time and spatial derivatives. To accurately simulate natural phenomena using this technology fine spatial and temporal discretizations are required, leading to large-scale linear systems or matrix equations, especially whenever more than one space dimension is considered. The discretization of fractional differential equations typically involves dense matrices with a Toeplitz or in the variable coefficient case Toeplitz-like structure . We combine the fast evaluation of Toeplitz matrices and their circulant preconditioners with state-of-the-art linear matrix equation methods to efficiently solve these problems, both in terms of CPU time and memory requirements. Additionally, we illustrate how these techniqes can be adapted when variable coefficients are present. Numerical experiments on typical differential problems with fractional derivatives in both space and time showing the effectiveness of the approaches are reported.

Spectral analysis and structure preserving preconditioners for fractional diffusion equations

Fractional partial order diffusion equations are a generalization of classical partial differential equations, used to model anomalous diffusion phenomena. When using the implicit Euler formula and the shifted Grunwald formula, it has been shown that the related discretizations lead to a linear system whose coefficient matrix has a Toeplitz-like structure. In this paper we focus our attention on the case of variable diffusion coefficients. Under appropriate conditions, we show that the sequence of the coefficient matrices belongs to the Generalized Locally Toeplitz class and we compute the symbol describing its asymptotic eigenvalue/singular value distribution, as the matrix size diverges. We employ the spectral information for analyzing known methods of preconditioned Krylov and multigrid type, with both positive and negative results and with a look forward to the multidimensional setting. We also propose two new tridiagonal structure preserving preconditioners to solve the resulting linear system, with Krylov methods such as CGNR and GMRES. A number of numerical examples show that our proposal is more effective than recently used circulant preconditioners.

Convergence of the scheme with weights for the numerical solution of a heat conduction equation with delay for the case of variable coefficient of heat conductivity

One-dimensional parabolic equations with delay effects in the time component for the case of variable coefficient of heat conductivity are considered. The scheme with weights is constructed for the numerical solution of these equations. The order of approximation error for the constructed scheme, stability, and order of convergence are investigated.