arXiv subjects

Dmitriy Leykekhman

Publications and source records attributed to Dmitriy Leykekhman.

At least 19 recordsLinked to original sources

arXiv2025

Fully discrete error analysis of finite element discretizations of time-dependent Stokes equations in a stream-function formulation

In this paper we establish best approximation type error estimates for the fully discrete Galerkin solutions of the time-dependent Stokes problem using the stream-function formulation. For the time discretization we use the discontinuous Galerkin method of arbitrary degree, whereas we present the space discretization in a general framework. This makes our result applicable for a wide variety of space discretization methods, provided some Galerkin orthogonality conditions are satisfied. As an example, conformal $C^1$ and $C^0$ interior penalty methods are covered by our analysis. The results do not require any additional regularity assumptions beyond the natural regularity given by the domain and data and can be used for optimal control problems.

math.NA↗

arXiv2025

The dimension spectrum of the infinitely generated Apollonian gasket

We prove that the infinitely generated Apollonian gasket has full Hausdorff dimension spectrum. Our proof, which is computer assisted, relies on an iterative technique introduced by the first three authors in [3] and on a flexible method for rigorously estimating Hausdorff dimensions of limit sets of conformal iterated function systems, which we recently developed in [5]. Another key ingredient in our proof is obtaining reasonably sized distortion constants for the (infinite) Apollonian iterated function system.

math.DS↗

arXiv2024

Rigorous Hausdorff dimension estimates for conformal fractals

We develop a versatile framework which allows us to rigorously estimate the Hausdorff dimension of maximal conformal graph directed Markov systems in $\mathbb{R}^n$ for $n \geq 2$. Our method is based on piecewise linear approximations of the eigenfunctions of the Perron-Frobenius operator via a finite element framework for discretization and iterative mesh schemes. One key element in our approach is obtaining bounds for the derivatives of these eigenfunctions, which, besides being essential for the implementation of our method, are of independent interest.

math.DS↗

arXiv2024

A priori error estimates for optimal control problems governed by the transient Stokes equations and subject to state constraints pointwise in time

In this paper, we consider a state constrained optimal control problem governed by the transient Stokes equations. The state constraint is given by an L2 functional in space, which is required to fulfill a pointwise bound in time. The discretization scheme for the Stokes equations consists of inf-sup stable finite elements in space and a discontinuous Galerkin method in time, for which we have recently established best approximation type error estimates. Using these error estimates, for the discrete control problem we establish error estimates and as a by-product we show an improved regularity for the optimal control. We complement our theoretical analysis with numerical results.

math.NA↗

arXiv2024

Weak maximum principle of finite element methods for parabolic equations in polygonal domains

The weak maximum principle of finite element methods for parabolic equations is proved for both semi-discretization in space and fully discrete methods with $k$-step backward differentiation formulae for $k = 1,... ,6$, on a two-dimensional general polygonal domain or a three-dimensional convex polyhedral domain. The semi-discrete result is established via a dyadic decomposition argument and local energy estimates in which the nonsmoothness of the domain can be handled. The fully discrete result for multistep backward differentiation formulae is proved by utilizing the solution representation via the discrete Laplace transform and the resolvent estimates, which are inspired by the analysis of convolutional quadrature for parabolic and fractional-order partial differential equations.

math.NA↗

arXiv2023

$L^2(I;H^1(Ω))$ and $L^2(I;L^2(Ω))$ best approximation type error estimates for Galerkin solutions of transient Stokes problems

In this paper we establish best approximation type estimates for the fully discrete Galerkin solutions of transient Stokes problem in $L^2(I;L^2(Ω)^d)$ and $L^2(I;H^1(Ω)^d)$ norms. These estimates fill the gap in the error analysis of the transient Stokes problems and have a number of applications. The analysis naturally extends to inhomogeneous parabolic problems. The best type $L^2(I;H^1(Ω))$ error estimates seems to be new even for scalar parabolic problems.

math.NA↗

arXiv2023

Fully Discrete Pointwise Smoothing Error Estimates for Measure Valued Initial Data

In this paper we analyze a homogeneous parabolic problem with initial data in the space of regular Borel measures. The problem is discretized in time with a discontinuous Galerkin scheme of arbitrary degree and in space with continuous finite elements of orders one or two. We show parabolic smoothing results for the continuous, semidiscrete and fully discrete problems. Our main results are interior $L^\infty$ error estimates for the evaluation at the endtime, in cases where the initial data is supported in a subdomain. In order to obtain these, we additionally show interior $L^\infty$ error estimates for $L^2$ initial data and quadratic finite elements, which extends the corresponding result previously established by the authors for linear finite elements.

math.NA↗

arXiv2022

Local energy estimates for the fractional Laplacian

The integral fractional Laplacian of order $s \in (0,1)$ is a nonlocal operator. It is known that solutions to the Dirichlet problem involving such an operator exhibit an algebraic boundary singularity regardless of the domain regularity. This, in turn, deteriorates the global regularity of solutions and as a result the global convergence rate of the numerical solutions. For finite element discretizations, we derive local error estimates in the $H^s$-seminorm and show optimal convergence rates in the interior of the domain by only assuming meshes to be shape-regular. These estimates quantify the fact that the reduced approximation error is concentrated near the boundary of the domain. We illustrate our theoretical results with several numerical examples.

math.NA↗

arXiv2021

Fully discrete best approximation type estimates in $L^{\infty}(I;L^2(Ω)^d)$ for finite element discretizations of the transient Stokes equations

In this article we obtain an optimal best approximation type result for fully discrete approximations of the transient Stokes problem. For the time discretization we use the discontinuous Galerkin method and for the spatial discretization we use standard finite elements for the Stokes problem satisfying the discrete inf-sup condition. The analysis uses the technique of discrete maximal parabolic regularity. The results require only natural assumptions on the data and do not assume any additional smoothness of the solutions.

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arXiv2020

Weak discrete maximum principle of finite element methods in convex polyhedra

We prove that the Galerkin finite element solution $u_h$ of the Laplace equation in a convex polyhedron $\varOmega$, with a quasi-uniform tetrahedral partition of the domain and with finite elements of polynomial degree $r\ge 1$, satisfies the following weak maximum principle: \begin{align*} \left\|u_{h}\right\|_{L^{\infty}(\varOmega)} \le C\left\|u_{h}\right\|_{L^{\infty}(\partial \varOmega)} , \end{align*} with a constant $C$ independent of the mesh size $h$. By using this result, we show that the Ritz projection operator $R_h$ is stable in $L^\infty$ norm uniformly in $h$ for $r\geq 2$, i.e. \begin{align*} \|R_hu\|_{L^{\infty}(\varOmega)} \le C\|u\|_{L^{\infty}(\varOmega)} . \end{align*} Thus we remove a logarithmic factor appearing in the previous results for convex polyhedral domains.

math.NA↗

arXiv2019

Global and local pointwise error estimates for finite element approximations to the Stokes problem on convex polyhedra

The main goal of the paper is to show new stability and localization results for the finite element solution of the Stokes system in $W^{1,\infty}$ and $L^{\infty}$ norms under standard assumptions on the finite element spaces on quasi-uniform meshes in two and three dimensions. Although interior error estimates are well-developed for the elliptic problem, they appear to be new for the Stokes system on unstructured meshes. To obtain these results we extend previously known stability estimates for the Stokes system using regularized Green's functions.

math.NA↗

arXiv2019

Numerical Analysis of Sparse Initial Data Identification for Parabolic Problems

In this paper we consider a problem of initial data identification from the final time observation for homogeneous parabolic problems. It is well-known that such problems are exponentially ill-posed due to the strong smoothing property of parabolic equations. We are interested in a situation when the initial data we intend to recover is known to be sparse, i.e. its support has Lebesgue measure zero. We formulate the problem as an optimal control problem and incorporate the information on the sparsity of the unknown initial data into the structure of the objective functional. In particular, we are looking for the control variable in the space of regular Borel measures and use the corresponding norm as a regularization term in the objective functional. This leads to a convex but non-smooth optimization problem. For the discretization we use continuous piecewise linear finite elements in space and discontinuous Galerkin finite elements of arbitrary degree in time. For the general case we establish error estimates for the state variable. Under a certain structural assumption, we show that the control variable consists of a finite linear combination of Dirac measures. For this case we obtain error estimates for the locations of Dirac measures as well as for the corresponding coefficients. The key to the numerical analysis are the sharp smoothing type pointwise finite element error estimates for homogeneous parabolic problems, which are of independent interest. Moreover, we discuss an efficient algorithmic approach to the problem and show several numerical experiments illustrating our theoretical results.

math.OC↗

arXiv2018

On the dimension spectrum of infinite subsystems of continued fractions

In this paper we study the dimension spectrum of continued fractions with coefficients restricted to infinite subsets of natural numbers. We prove that if $E$ is any arithmetic progression, the set of primes, or the set of squares $\{n^2\}_{n \in \mathbb{N}}$, then the continued fractions whose digits lie in $E$ have full dimension spectrum, which we denote by $DS(\mathcal{CF}_E)$. Moreover we prove that if $E$ is an infinite set of consecutive powers then the dimension spectrum $DS(\mathcal{CF}_E)$ always contains a non trivial interval. We also show that there exists some $E \subset \mathbb{N}$ and two non-trivial intervals $I_1, I_2$, such that $DS(\mathcal{CF}_E) \cap I_1=I_1$ and $DS(\mathcal{CF}_E) \cap I_2$ is a Cantor set. On the way we employ the computational approach of Falk and Nussbaum in order to obtain rigorous effective estimates for the Hausdorff dimension of continued fractions whose entries are restricted to infinite sets.

math.DS↗

arXiv2018

The dimension spectrum of graph directed Markov systems

In this paper we study the dimension spectrum of general conformal graph directed Markov systems modeled by countable state symbolic subshifts of finite type. We perform a comprehensive study of the dimension spectrum addressing questions regarding its size and topological structure. As a corollary we obtain that the dimension spectrum of infinite conformal iterated function systems is compact and perfect. On the way we revisit the role of the parameter $θ$ in graph directed Markov systems and we show that new phenomena arise. We also establish topological pressure estimates for subsystems in the abstract setting of symbolic dynamics with countable alphabets. These estimates play a crucial role in our proofs regarding the dimension spectrum, and they allow us to study Hausdorff dimension asymptotics for subsystems. Finally we narrow our focus to the dimension spectrum of conformal iterated function systems and we prove, among other things, that the iterated function system resulting from the complex continued fractions algorithm has full dimension spectrum. We thus give a positive answer to the Texan conjecture for complex continued fractions.

math.DS↗

arXiv2017

Discrete maximal parabolic regularity for Galerkin finite element methods for non-autonomous parabolic problem s

The main goal of the paper is to establish time semidiscrete and space-time fully discrete maximal parabolic regularity for the lowest order time discontinuous Galerkin solution of linear parabolic equations with time-dependent coefficients. Such estimates have many applications. As one of the applications we establish best approximations type results with respect to the $L^p(0,T;L^2(\Omega))$ norm for $1\le p\le \infty$.

math.NA↗

arXiv2017

Optimal a priori error estimates of parabolic optimal control problems with a moving point control

In this paper we consider a parabolic optimal control problem with a Dirac type control with moving point source in two space dimensions. We discretize the problem with piecewise constant functions in time and continuous piecewise linear finite elements in space. For this discretization we show optimal order of convergence with respect to the time and the space discretization parameters modulo some logarithmic terms. Error analysis for the same problem was carried out in the recent paper [17], however, the analysis there contains a serious flaw. One of the main goals of this paper is to provide the correct proof. The main ingredients of our analysis are the global and local error estimates on a curve, that have an independent interest.

math.NA↗

arXiv2016

Global and interior pointwise best approximation results for the gradient of Galerkin solutions for parabolic problems

In this paper we establish best approximation property of fully discrete Galerkin solutions of second order parabolic problems on convex polygonal and polyhedral domains in the $L^\infty(I;W^{1,\infty}(\Om))$ norm. The discretization method consists of continuous Lagrange finite elements in space and discontinuous Galerkin methods of arbitrary order in time. The method of the proof differs from the established fully discrete error estimate techniques and uses only elliptic results and discrete maximal parabolic regularity for discontinuous Galerkin methods established by the authors in \cite{LeykekhmanD_VexlerB_2016b}. In addition, the proof does not require any relationship between spatial mesh sizes and time steps. We also establish interior best approximation property that shows more local dependence of the error at a point.

math.NA↗

arXiv2015

Pointwise best approximation results for Galerkin finite element solutions of parabolic problems

In this paper we establish a best approximation property of fully discrete Galerkin finite element solutions of second order parabolic problems on convex polygonal and polyhedral domains in the $L^\infty$ norm. The discretization method uses of continuous Lagrange finite elements in space and discontinuous Galerkin methods in time of an arbitrary order. The method of proof differs from the established fully discrete error estimate techniques and for the first time allows to obtain such results in three space dimensions. It uses elliptic results, discrete resolvent estimates in weighted norms, and the discrete maximal parabolic regularity for discontinuous Galerkin methods established by the authors in [16]. In addition, the proof does not require any relationship between spatial mesh sizes and time steps. We also establish a local best approximation property that shows a more local behavior of the error at a given point.

math.NA↗