arXiv subjects

Raytcho Lazarov

Publications and source records attributed to Raytcho Lazarov.

26 records · Page 2Linked to original sources

arXiv2014

The Galerkin Finite Element Method for A Multi-term Time-Fractional Diffusion equation

We consider the initial/boundary value problem for a diffusion equation involving multiple time-fractional derivatives on a bounded convex polyhedral domain. We analyze a space semidiscrete scheme based on the standard Galerkin finite element method using continuous piecewise linear functions. Nearly optimal error estimates for both cases of initial data and inhomogeneous term are derived, which cover both smooth and nonsmooth data. Further we develop a fully discrete scheme based on a finite difference discretization of the time-fractional derivatives, and discuss its stability and error estimate. Extensive numerical experiments for one and two-dimension problems confirm the convergence rates of the theoretical results.

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arXiv2013

A multiscale HDG method for second order elliptic equations. Part I. Polynomial and homogenization-based multiscale spaces

We introduce a finite element method for numerical upscaling of second order elliptic equations with highly heterogeneous coefficients. The method is based on a mixed formulation of the problem and the concepts of the domain decomposition and the hybrid discontinuous Galerkin methods. The method utilizes three different scales: (1) the scale of the partition of the domain of the problem, (2) the sale of partition of the boundaries of the subdomains (related to the corresponding space of Lagrange multipliers), and (3) the fine grid scale that is assumed to resolve the scale of the heterogeneous variation of the coefficients. Our proposed method gives a flexible framework that (1) couples independently generated multiscale basis functions in each coarse patch (2) provides a stable global coupling independent of local discretization, physical scales and contrast (3) allows avoiding any constraints on coarse spaces. In this paper, we develop and study a multiscale HDG method that uses polynomial and homogenization-based multiscale spaces. These coarse spaces are designed for problems with scale separation. In our consequent paper, we plan to extend our flexible HDG framework to more challenging multiscale problems with non-separable scales and high contrast and consider enriched coarse spaces that use appropriate local spectral problems.

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arXiv2013

Error Analysis of Finite Element Methods for Space-Fractional Parabolic Equations

We consider an initial/boundary value problem for one-dimensional fractional-order parabolic equations with a space fractional derivative of Riemann-Liouville type and order $α\in (1,2)$. We study a spatial semidiscrete scheme with the standard Galerkin finite element method with piecewise linear finite elements, as well as fully discrete schemes based on the backward Euler method and Crank-Nicolson method. Error estimates in the $L^2\II$- and $H^{α/2}\II$-norm are derived for the semidiscrete scheme, and in the $L^2\II$-norm for the fully discrete schemes. These estimates are for both smooth and nonsmooth initial data, and are expressed directly in terms of the smoothness of the initial data. Extensive numerical results are presented to illustrate the theoretical results.

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arXiv2013

A Finite Element Method for the Fractional Sturm-Liouville Problem

In this work, we propose an efficient finite element method for solving fractional Sturm-Liouville problems involving either the Caputo or Riemann-Liouville derivative of order $α\in(1,2)$ on the unit interval $(0,1)$. It is based on novel variational formulations of the eigenvalue problem. Error estimates are provided for the finite element approximations of the eigenvalues. Numerical results are presented to illustrate the efficiency and accuracy of the method. The results indicate that the method can achieve a second-order convergence for both fractional derivatives, and can provide accurate approximations to multiple eigenvalues simultaneously.

math.NA↗

arXiv2013

Variational formulation of problems involving fractional order differential operators

In this work, we consider boundary value problems involving Caputo and Riemann-Liouville fractional derivatives of order $α\in(1,2)$ on the unit interval $(0,1)$. These fractional derivatives lead to non-symmetric boundary value problems, which are investigated from a variational point of view. The variational problem for the Riemann-Liouville case is coercive on the space $H_0^{α/2}(0,1)$ but the solutions are less regular, whereas that for the Caputo case involves different test and trial spaces. The numerical analysis of these problems requires the so-called shift theorems which show that the solutions of the variational problem are more regular. The regularity pickup enables one to establish convergence rates of the finite element approximations. Finally, numerical results are presented to illustrate the error estimates.

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arXiv2013

Error Analysis of Semidiscrete Finite Element Methods for Inhomogeneous Time-Fractional Diffusion

We consider the initial boundary value problem for the inhomogeneous time-fractional diffusion equation with a homogeneous Dirichlet boundary condition and a nonsmooth right hand side data in a bounded convex polyhedral domain. We analyze two semidiscrete schemes based on the standard Galerkin and lumped mass finite element methods. Almost optimal error estimates are obtained for right hand side data $f(x,t)\in L^\infty(0,T;\dot H^q(Ω))$, $-1< q \le 1$, for both semidiscrete schemes. For lumped mass method, the optimal $L^2(Ω)$-norm error estimate requires symmetric meshes. Finally, numerical experiments for one- and two-dimensional examples are presented to verify our theoretical results.

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arXiv2013

Galerkin FEM for fractional order parabolic equations with initial data in $H^{-s},~0 < s \le 1$

We investigate semi-discrete numerical schemes based on the standard Galerkin and lumped mass Galerkin finite element methods for an initial-boundary value problem for homogeneous fractional diffusion problems with non-smooth initial data. We assume that $Ω\subset \mathbb{R}^d$, $d=1,2,3$ is a convex polygonal (polyhedral) domain. We theoretically justify optimal order error estimates in $L_2$- and $H^1$-norms for initial data in $H^{-s}(Ω),~0\le s \le 1$. We confirm our theoretical findings with a number of numerical tests that include initial data $v$ being a Dirac $δ$-function supported on a $(d-1)$-dimensional manifold.

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arXiv2012

Error estimates for a semidiscrete finite element method for fractional order parabolic equations

We consider the initial boundary value problem for the homogeneous time-fractional diffusion equation $\partial^α_t u - \De u =0$ ($0< α< 1$) with initial condition $u(x,0)=v(x)$ and a homogeneous Dirichlet boundary condition in a bounded polygonal domain $Ω$. We shall study two semidiscrete approximation schemes, i.e., Galerkin FEM and lumped mass Galerkin FEM, by using piecewise linear functions. We establish optimal with respect to the regularity of the solution error estimates, including the case of nonsmooth initial data, i.e., $v \in L_2(Ω)$.

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