arXiv subjects

Endre Süli

Publications and source records attributed to Endre Süli.

At least 37 records · Page 2Linked to original sources

arXiv2022

Existence of Large-Data Global Weak Solutions to Kinetic Models of Nonhomogeneous Dilute Polymeric Fluids

We prove the existence of large-data global-in-time weak solutions to a general class of coupled bead-spring chain models with finitely extensible nonlinear elastic (FENE) type spring potentials for nonhomogeneous incompressible dilute polymeric fluids in a bounded domain in $\mathbb{R}^d$, $d=2$ or $3$. The class of models under consideration involves the Navier--Stokes system with variable density, where the viscosity coefficient depends on both the density and the polymer number density, coupled to a Fokker--Planck equation with a density-dependent drag coefficient. The proof is based on combining a truncation of the probability density function with a two-stage Galerkin approximation and weak compactness and compensated compactness techniques to pass to the limits in the sequence of Galerkin approximations and in the truncation level.

math.AP↗

arXiv2021

Finite element appoximation and augmented Lagrangian preconditioning for anisothermal implicitly-constituted non-Newtonian flow

We devise 3-field and 4-field finite element approximations of a system describing the steady state of an incompressible heat-conducting fluid with implicit non-Newtonian rheology. We prove that the sequence of numerical approximations converges to a weak solution of the problem. We develop a block preconditioner based on augmented Lagrangian stabilisation for a discretisation based on the Scott-Vogelius finite element pair for the velocity and pressure. The preconditioner involves a specialised multigrid algorithm that makes use of a space-decomposition that captures the kernel of the divergence and non-standard intergrid transfer operators. The preconditioner exhibits robust convergence behaviour when applied to the Navier-Stokes and power-law systems, including temperature-dependent viscosity, heat conductivity and viscous dissipation.

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arXiv2021

Adaptive iterative linearised finite element methods for implicitly constituted incompressible fluid flow problems and its application to Bingham fluids

In this work, we introduce an iterative linearised finite element method for the solution of Bingham fluid flow problems. The proposed algorithm has the favourable property that a subsequence of the sequence of iterates generated converges weakly to a solution of the problem. This will be illustrated by two numerical experiments.

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arXiv2021

On the convergence rate of the Kačanov scheme for shear-thinning fluids

We explore the convergence rate of the Kačanov iteration scheme for different models of shear-thinning fluids, including Carreau and power-law type explicit quasi-Newtonian constitutive laws. It is shown that the energy difference contracts along the sequence generated by the iteration. In addition, an a posteriori computable contraction factor is proposed, which improves, on finite-dimensional Galerkin spaces, previously derived bounds on the contraction factor in the context of the power-law model. Significantly, this factor is shown to be independent of the choice of the cut-off parameters whose use was proposed in the literature for the Kačanov iteration applied to the power-law model. Our analytical findings are confirmed by a series of numerical experiments.

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arXiv2021

Finite Element Approximation of Steady Flows of Colloidal Solutions

We consider the mathematical analysis and numerical approximation of a system of nonlinear partial differential equations that arises in models that have relevance to steady isochoric flows of colloidal suspensions. The symmetric velocity gradient is assumed to be a monotone nonlinear function of the deviatoric part of the Cauchy stress tensor. We prove the existence of a unique weak solution to the problem, and under the additional assumption that the nonlinearity involved in the constitutive relation is Lipschitz continuous we also prove uniqueness of the weak solution. We then construct mixed finite element approximations of the system using both conforming and nonconforming finite element spaces. For both of these we prove the convergence of the method to the unique weak solution of the problem, and in the case of the conforming method we provide a bound on the error between the analytical solution and its finite element approximation in terms of the best approximation error from the finite element spaces. We propose first a Lions-Mercier type iterative method and next a classical fixed-point algorithm to solve the finite-dimensional problems resulting from the finite element discretisation of the system of nonlinear partial differential equations under consideration and present numerical experiments that illustrate the practical performance of the proposed numerical method.

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arXiv2021

Uniform Hölder-norm bounds for finite element approximations of second-order elliptic equations

We develop a discrete counterpart of the De Giorgi-Nash-Moser theory, which provides uniform Hölder-norm bounds on continuous piecewise affine finite element approximations of second-order linear elliptic problems of the form $-\nabla \cdot(A\nabla u)=f-\nabla\cdot F$ with $A\in L^\infty(Ω;\mathbb{R}^{n\times n})$ a uniformly elliptic matrix-valued function, $f\in L^{q}(Ω)$, $F\in L^p(Ω;\mathbb{R}^n)$, with $p > n$ and $q > n/2$, on $A$-nonobtuse shape-regular triangulations, which are not required to be quasi-uniform, of a bounded polyhedral Lipschitz domain $Ω\subset \mathbb{R}^n$.

math.NA↗

arXiv2021

A conservative fully-discrete numerical method for the regularised shallow water wave equations

The paper proposes a new, conservative fully-discrete scheme for the numerical solution of the regularised shallow water Boussinesq system of equations in the cases of periodic and reflective boundary conditions. The particular system is one of a class of equations derived recently and can be used in practical simulations to describe the propagation of weakly nonlinear and weakly dispersive long water waves, such as tsunamis. Studies of small-amplitude long waves usually require long-time simulations in order to investigate scenarios such as the overtaking collision of two solitary waves or the propagation of transoceanic tsunamis. For long-time simulations of non-dissipative waves such as solitary waves, the preservation of the total energy by the numerical method can be crucial in the quality of the approximation. The new conservative fully-discrete method consists of a Galerkin finite element method for spatial semidiscretisation and an explicit relaxation Runge--Kutta scheme for integration in time. The Galerkin method is expressed and implemented in the framework of mixed finite element methods. The paper provides an extended experimental study of the accuracy and convergence properties of the new numerical method. The experiments reveal a new convergence pattern compared to standard Galerkin methods.

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arXiv2020

Existence and uniqueness of global weak solutions to strain-limiting viscoelasticity with Dirichlet boundary data

We consider a system of evolutionary equations that is capable of describing certain viscoelastic effects in linearized yet nonlinear models of solid mechanics. The essence of the paper is that the constitutive relation, involving the Cauchy stress, the small strain tensor and the symmetric velocity gradient, is given in an implicit form. For a large class of implicit constitutive relations we establish the existence and uniqueness of a global-in-time large-data weak solution. We then focus on the class of so-called limiting strain models, i.e., models for which the magnitude of the strain tensor is known to remain small a priori, regardless of the magnitude of the Cauchy stress tensor. For this class of models, a new technical difficulty arises, which is that the Cauchy stress is only an integrable function over its domain of definition, resulting in the underlying function spaces being nonreflexive and thus the weak compactness of bounded sequences of elements of these spaces is lost. Nevertheless, even for problems of this type we are able to provide a satisfactory existence theory, as long as the initial data have finite elastic energy and the boundary data fulfill natural compatibility conditions.

math.AP↗

arXiv2020

Existence of large-data global weak solutions to a model of a strain-limiting viscoelastic body

We prove the existence of a unique large-data global-in-time weak solution to a class of models of the form $\mathbf{u}_{tt} = \mathrm{div}(\mathbb{T}) + \mathbf{f}$ for viscoelastic bodies exhibiting strain-limiting behaviour, where the constitutive equation, relating the linearised strain tensor $\boldsymbolε(\mathbf{u})$ to the Cauchy stress tensor $\mathbb{T}$, is assumed to be of the form $\boldsymbolε(\mathbf{u}_t) +α\boldsymbolε(\mathbf{u})= F(\mathbb{T})$, where we define $F(\mathbb{T}) = (1 + |\mathbb{T}|^a)^{-\frac{1}{a}}\mathbb{T}$, for constant parameters $α\in (0, \infty)$ and $a\in (0, \infty)$, in any number $d$ of space dimensions, with periodic boundary conditions. The Cauchy stress $\mathbb{T}$ is show to belong to $L^1(Q)^{d\times d}$ over the space-time domain $Q$. In particular, in three space dimensions, if $a\in (0, \frac{2}{7})$, then in fact $\mathbb{T}\in L^{1+δ}(Q)^{d\times d}$ for a $δ>0$, the value of which depends only on $a$.

math.AP↗

arXiv2020

Mixed finite element approximation of periodic Hamilton--Jacobi--Bellman problems with application to numerical homogenization

In the first part of the paper, we propose and rigorously analyze a mixed finite element method for the approximation of the periodic strong solution to the fully nonlinear second-order Hamilton--Jacobi--Bellman equation with coefficients satisfying the Cordes condition. These problems arise as the corrector problems in the homogenization of Hamilton--Jacobi--Bellman equations. The second part of the paper focuses on the numerical homogenization of such equations, more precisely on the numerical approximation of the effective Hamiltonian. Numerical experiments demonstrate the approximation scheme for the effective Hamiltonian and the numerical solution of the homogenized problem.

math.NA↗

arXiv2020

On incompressible heat-conducting viscoelastic rate-type fluids with stress-diffusion and purely spherical elastic response

We prove the existence of large-data global-in-time weak solutions to an evolutionary PDE system describing flows of incompressible \emph{heat-conducting} viscoelastic rate-type fluids with stress-diffusion, subject to a stick-slip boundary condition for the velocity and a homogeneous Neumann boundary condition for the extra stress tensor. In the introductory section we develop the thermodynamic foundations of the proposed model, and we document the role of thermodynamics in obtaining critical structural relations between the quantities of interest. These structural relations are then exploited in the mathematical analysis of the governing equations. In particular, the definition of weak solution is motivated by the thermodynamic basis of the model. The extra stress tensor describing the elastic response of the fluid is in our case purely spherical, which is a simplification from the physical point of view. The model nevertheless exhibits features that require novel mathematical ideas in order to deal with the technically complex structure of the associated internal energy and the more complicated forms of the corresponding entropy and energy fluxes. The paper provides the first rigorous proof of the existence of large-data global-in-time weak solutions to the governing equations for \emph{coupled thermo-mechanical processes} in viscoelastic rate-type fluids.

math.AP↗

arXiv2020

Finite Element Approximation of a Strain-Limiting Elastic Model

We construct a finite element approximation of a strain-limiting elastic model on a bounded open domain in $\mathbb{R}^d$, $d \in \{2,3\}$. The sequence of finite element approximations is shown to exhibit strong convergence to the unique weak solution of the model. Assuming that the material parameters featuring in the model are Lipschitz-continuous, and assuming that the weak solution has additional regularity, the sequence of finite element approximations is shown to converge with a rate. An iterative algorithm is constructed for the solution of the system of nonlinear algebraic equations that arises from the finite element approximation. An appealing feature of the iterative algorithm is that it decouples the monotone and linear elastic parts of the nonlinearity in the model. In particular, our choice of piecewise constant approximation for the stress tensor (and continuous piecewise linear approximation for the displacement) allows us to compute the monotone part of the nonlinearity by solving an algebraic system with $d(d+1)/2$ unknowns independently on each element in the subdivision of the computational domain. The theoretical results are illustrated by numerical experiments.

math.NA↗

arXiv2019

Numerical Analysis of Unsteady Implicitly Constituted Incompressible Fluids: Three-Field Formulation

In the classical theory of fluid mechanics a linear relationship between the shear stress and the symmetric velocity gradient tensor is often assumed. Even when a nonlinear relationship is assumed, it is typically formulated in terms of an explicit relation. Implicit constitutive models provide a theoretical framework that generalises this, allowing for general implicit constitutive relations. Since it is generally not possible to solve explicitly for the shear stress in the constitutive relation, a natural approach is to include the shear stress as a fundamental unknown in the formulation of the problem. In this work we present a mixed formulation with this feature, discuss its solvability and approximation using mixed finite element methods, and explore the convergence of the numerical approximations to a weak solution of the model.

math.NA↗

arXiv2019

Gamma-convergence of a shearlet-based Ginzburg--Landau energy

We introduce two shearlet-based Ginzburg--Landau energies, based on the continuous and the discrete shearlet transform. The energies result from replacing the elastic energy term of a classical Ginzburg--Landau energy by the weighted $L^2$-norm of a shearlet transform. The asymptotic behaviour of sequences of these energies is analysed within the framework of $Γ$-convergence and the limit energy is identified. We show that the limit energy of a characteristic function is an anisotropic surface integral over the interfaces of that function. We demonstrate that the anisotropy of the limit energy can be controlled by weighting the underlying shearlet transforms according to their directional parameter.

math.FA↗

arXiv2019

Well-posedness of the fractional Zener wave equation for heterogenous viscoelastic materials

We explore the well-posedness of the fractional version of Zener's wave equation for viscoelastic solids, which is based on a constitutive law relating the stress tensor $\boldsymbolσ$ to the strain tensor $\boldsymbol\varepsilon(\bf u)$, with $\bf u$ being the displacement vector, defined by: $(1+τD_t^α) {\boldsymbolσ}=(1+ρD_t^α)[2μ{\boldsymbol\varepsilon}({\bf u})+λ\text{tr}(\boldsymbol\varepsilon(\bf u)) \bf ]$. Here $μ,λ\in\mathrm{L}^\infty(Ω)$, $μ$ is the shear modulus bounded below by a positive constant, and $λ\geq 0$ is first Lamé coefficient, $D_t^α$, with $α\in (0,1)$, is the Caputo time-derivative, $τ>0$ is the characteristic relaxation time and $ρ\geqτ$ is the characteristic retardation time. We show that, when coupled with the equation of motion $\varrho \ddot{\bf u} = \text{Div}{\boldsymbolσ} + \bf F$, considered in a bounded open Lipschitz domain $Ω$ in $\mathbb{R}^3$ and over a time interval $(0,T]$, where $\varrho\in \mathrm{L}^\infty(Ω)$ is the density of the material, bounded below by a positive constant, and $\bf F$ is a specified load vector, the resulting model is well-posed in the sense that the associated initial-boundary-value problem, with initial conditions ${\bf u}(0,\mathbf{x}) = {\bf g}(\mathbf{x})$, $\dot{\bf u}(0,\mathbf{x}) = \bf h(\mathbf{x})$, ${\boldsymbolσ}(0,\mathbf{x}) = {\bf s}(\mathbf{x})$, for $\mathbf{x} \in Ω$, and a homogeneous Dirichlet boundary condition, possesses a unique weak solution for any choice of ${\bf g }\in [\mathrm{H}^1_0(Ω)]^3$, ${\bf h}\in [\mathrm{L}^2(Ω)]^3$, and ${\bf S} = {\bf S}^{\rm T} \in [\mathrm{L}^2(Ω)]^{3 \times 3}$, and any load vector ${\bf F} \in\mathrm{L}^2(0,T;[\mathrm{L}^2(Ω)]^3)$, and that this unique weak solution depends continuously on the initial data and the load vector.

math.AP↗

arXiv2019

The incompressible limit of compressible finitely extensible nonlinear bead-spring chain models for dilute polymeric fluids

We explore the behaviour of global-in-time weak solutions to a class of bead-spring chain models, with finitely extensible nonlinear elastic (FENE) spring potentials, for dilute polymeric fluids. In the models under consideration the solvent is assumed to be a compressible, isentropic, viscous, isothermal Newtonian fluid, confined to a bounded open domain in $\mathbb{R}^3$, and the velocity field is assumed to satisfy a complete slip boundary condition. We show that as the Mach number tends to zero the system is driven to its incompressible counterpart.

math.AP↗

arXiv2019

Finite Element Approximation of Elliptic Homogenization Problems in Nondivergence-Form

We use uniform $W^{2,p}$ estimates to obtain corrector results for periodic homogenization problems of the form $A(x/\varepsilon):D^2 u_{\varepsilon} = f$ subject to a homogeneous Dirichlet boundary condition. We propose and rigorously analyze a numerical scheme based on finite element approximations for such nondivergence-form homogenization problems. The second part of the paper focuses on the approximation of the corrector and numerical homogenization for the case of nonuniformly oscillating coefficients. Numerical experiments demonstrate the performance of the scheme.

math.NA↗

arXiv2019

Optimal order finite difference approximation of generalized solutions to the biharmonic equation in a cube

We prove an optimal order error bound in the discrete $H^2(Ω)$ norm for finite difference approximations of the first boundary-value problem for the biharmonic equation in $n$ space dimensions, with $n \in \{2,\dots,7\}$, whose generalized solution belongs to the Sobolev space $H^s(Ω) \cap H^2_0(Ω)$, for $\frac{1}{2} \max(5,n) < s \leq 4$, where $Ω= (0,1)^n$. The result extends the range of the Sobolev index $s$ in the best convergence results currently available in the literature to the maximal range admitted by the Sobolev embedding of $H^s(Ω)$ into $C(\overlineΩ)$ in $n$ space dimensions.

math.NA↗