Endre Süli

Lars Diening, Christian Kreuzer, Endre Süli

We develop the analysis of finite element approximations of implicit power-law-like models for viscous incompressible fluids. The Cauchy stress and the symmetric part of the velocity gradient in the class of models under consideration are related by a, possibly multi--valued, maximal monotone $r$-graph, with $1<r<\infty$. Using a variety of weak compactness techniques, including Chacon's biting lemma and Young measures, we show that a subsequence of the sequence of finite element solutions converges to a weak solution of the problem as the finite element discretization parameter $h$ tends to 0. A key new technical tool in our analysis is a finite element counterpart of the Acerbi--Fusco Lipschitz truncation of Sobolev functions.

Wolfgang Dahmen, Ronald DeVore, Lars Grasedyck et al.

A recurring theme in attempts to break the curse of dimensionality in the numerical approximations of solutions to high-dimensional partial differential equations (PDEs) is to employ some form of sparse tensor approximation. Unfortunately, there are only a few results that quantify the possible advantages of such an approach. This paper introduces a class $Σ_n$ of functions, which can be written as a sum of rank-one tensors using a total of at most $n$ parameters and then uses this notion of sparsity to prove a regularity theorem for certain high-dimensional elliptic PDEs. It is shown, among other results, that whenever the right-hand side $f$ of the elliptic PDE can be approximated with a certain rate $\mathcal{O}(n^{-r})$ in the norm of ${\mathrm H}^{-1}$ by elements of $Σ_n$, then the solution $u$ can be approximated in ${\mathrm H}^1$ from $Σ_n$ to accuracy $\mathcal{O}(n^{-r'})$ for any $r'\in (0,r)$. Since these results require knowledge of the eigenbasis of the elliptic operator considered, we propose a second "basis-free" model of tensor sparsity and prove a regularity theorem for this second sparsity model as well. We then proceed to address the important question of the extent such regularity theorems translate into results on computational complexity. It is shown how this second model can be used to derive computational algorithms with performance that breaks the curse of dimensionality on certain model high-dimensional elliptic PDEs with tensor-sparse data.

Leonardo E. Figueroa, Endre Süli

We investigate the convergence of a nonlinear approximation method introduced by Ammar et al. (J. Non-Newtonian Fluid Mech. 139:153-176, 2006) for the numerical solution of high-dimensional Fokker-Planck equations featuring in Navier-Stokes-Fokker-Planck systems that arise in kinetic models of dilute polymers. In the case of Poisson's equation on a rectangular domain in R^2, subject to a homogeneous Dirichlet boundary condition, the mathematical analysis of the algorithm was carried out recently by Le Bris, Lelièvre and Maday (Const. Approx. 30:621-651, 2009), by exploiting its connection to greedy algorithms from nonlinear approximation theory, explored, for example, by DeVore and Temlyakov (Adv. Comput. Math. 5:173-187, 1996); hence, the variational version of the algorithm, based on the minimization of a sequence of Dirichlet energies, was shown to converge. Here, we extend the convergence analysis of the pure greedy and orthogonal greedy algorithms considered by Le Bris et al. to a technically more complicated situation, where the Laplace operator is replaced by an Ornstein-Uhlenbeck operator of the kind that appears in Fokker-Planck equations that arise in bead-spring chain type kinetic polymer models with finitely extensible nonlinear elastic potentials, posed on a high-dimensional Cartesian product configuration space D = D_1 x ... x D_N contained in R^(N d), where each set D_i, i = 1, ..., N, is a bounded open ball in R^d, d = 2, 3.

Yves Capdeboscq, Timo Sprekeler, Endre Süli

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.

Graham Baird, Endre Süli

This paper concerns the construction and analysis of a numerical scheme for a mixed discrete-continuous fragmentation equation. A finite volume scheme is developed, based on a conservative formulation of a truncated version of the equations. The approximate solutions provided by this scheme are first shown to display conservation of mass and preservation of nonnegativity. Then, by utilising a Dunford--Pettis style argument, the sequence of approximate solutions generated is shown, under given restrictions on the model and the mesh, to converge (weakly) in an appropriate $L_1$ space to a weak solution to the problem. Additionally, by applying the methods and theory of operator semigroups, we are further able to show that weak solutions to the problem are unique and necessarily classical (differentiable) solutions. Finally, numerical simulations are performed to investigate the performance of the scheme and assess its rate of convergence.

Seungchan Ko, Petra Pustejovská, Endre Süli

We consider a system of nonlinear partial differential equations modelling the steady motion of an incompressible non-Newtonian fluid, which is chemically reacting. The governing system consists of a steady convection-diffusion equation for the concentration and the generalized steady Navier-Stokes equations, where the viscosity coefficient is a power-law type function of the shear-rate, and the coupling between the equations results from the concentration-dependence of the power-law index. This system of nonlinear partial differential equations arises in mathematical models of the synovial fluid found in the cavities of moving joints. We construct a finite element approximation of the model and perform the mathematical analysis of the numerical method in the case of two space dimensions. Key technical tools include discrete counterparts of the Bogovski\uı operator, De Giorgi's regularity theorem in two dimensions, and the Acerbi-Fusco Lipschitz truncation of Sobolev functions, in function spaces with variable integrability exponents.

Endre Süli, Tabea Tscherpel

Implicit constitutive theory provides a very general framework for fluid flow models, including both Newtonian and generalized Newtonian fluids, where the Cauchy stress tensor and the rate of strain tensor are assumed to be related by an implicit relation associated with a maximal monotone graph. For incompressible unsteady flows of such fluids, subject to a homogeneous Dirichlet boundary condition on a Lipschitz polytopal domain $Ω\subset \mathbb{R}^d$, $d \in \{2,3\}$, we investigate a fully-discrete approximation scheme, using a spatial mixed finite element approximation combined with backward Euler time-stepping. We show convergence of a subsequence of approximate solutions, when the velocity field belongs to the space of solenoidal functions contained in $L^\infty(0,T;L^2(Ω)^d)\cap L^q(0,T;W^{1,q}_0(Ω)^d)$, provided that $q\in \big(\frac{2d}{d+2},\infty\big)$, which is the maximal range for $q$ with respect to existence of weak solutions. This is achieved by a technique based on splitting and regularizing, the use of a solenoidal parabolic Lipschitz truncation method, a local Minty-type monotonicity result, and various weak compactness results.

Christoph Reisinger, Endre Süli, Alan Whitley

We consider hypoelliptic Kolmogorov equations in $n+1$ spatial dimensions, with $n\geq 1$, where the differential operator in the first $n$ spatial variables featuring in the equation is second-order elliptic, and with respect to the $(n+1)$st spatial variable the equation contains a pure transport term only and is therefore first-order hyperbolic. If the two differential operators, in the first $n$ and in the $(n+1)$st co-ordinate directions, do not commute, we benefit from hypoelliptic regularization in time, and the solution for $t>0$ is smooth even for a Dirac initial datum prescribed at $t=0$. We study specifically the case where the coefficients depend only on the first $n$ variables. In that case, a Fourier transform in the last variable and standard central finite difference approximation in the other variables can be applied for the numerical solution. We prove second-order convergence in the spatial mesh size for the model hypoelliptic equation $\frac{\partial u}{\partial t} + x \frac{\partial u}{\partial y} = \frac{\partial^2 u}{\partial x^2}$ subject to the initial condition $u(x,y,0) = δ(x) δ(y)$, with $(x,y) \in \mathbb{R} \times\mathbb{R}$ and $t>0$, proposed by Kolmogorov, and for an extension with $n=2$. We also demonstrate exponential convergence of an approximation of the inverse Fourier transform based on the trapezium rule. Lastly, we apply the method to a PDE arising in mathematical finance, which models the distribution of the hedging error under a mis-specified derivative pricing model.

Christian Kreuzer, Endre Süli

We develop the a posteriori error analysis of finite element approximations of implicit power-law-like models for viscous incompressible fluids. The Cauchy stress and the symmetric part of the velocity gradient in the class of models under consideration are related by a, possibly multi--valued, maximal monotone $r$-graph, with $\frac{2d}{d+1}<r<\infty$. We establish upper and lower bounds on the finite element residual, as well as the local stability of the error bound. We then consider an adaptive finite element approximation of the problem, and, under suitable assumptions, we show the weak convergence of the adaptive algorithm to a weak solution of the boundary-value problem. The argument is based on a variety of weak compactness techniques, including Chacon's biting lemma and a finite element counterpart of the Acerbi--Fusco Lipschitz truncation of Sobolev functions, introduced by L. Diening, C. Kreuzer and E. Süli [Finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology. SIAM J. Numer. Anal., 51(2), 984--1015].

Miles Caddick, Endre Süli

This paper is concerned with the proof of existence and numerical approximation of large-data global-in-time Young measure solutions to initial-boundary-value problems for multidimensional nonlinear parabolic systems of forward-backward type of the form $\partial_t u - \mbox{div}(a(Du)) + Bu = F$, where $B \in \mathbb{R}^{m \times m}$, $Bv \cdot v \geq 0$ for all $v \in \mathbb{R}^m$, $F$ is an $m$-component vector-function defined on a bounded open Lipschitz domain $Ω\subset \mathbb{R}^n$, and $a$ is a locally Lipschitz mapping of the form $a(A)=K(A)A$, where $K\,:\, \mathbb{R}^{m \times n} \rightarrow \mathbb{R}$. The function $a$ may have a nonstandard growth rate, in the sense that it is permitted to have unequal lower and upper growth rates. Furthermore, $a$ is not assumed to be monotone, nor is it assumed to be the gradient of a potential. Problems of this type arise in mathematical models of the atmospheric boundary layer and fall beyond the scope of monotone operator theory. We develop a numerical algorithm for the approximate solution of problems in this class, and we prove the convergence of the algorithm to a Young measure solution of the system under consideration.

Stefan Müller, Florian Schweiger, Endre Süli

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.

Alpha Albert Lee, Andreas Münch, Endre Süli

In this work, the sharp interface limit of the degenerate Cahn-Hilliard equation (in two space dimensions) with a polynomial double well free energy and a quadratic mobility is derived via a matched asymptotic analysis involving exponentially large and small terms and multiple inner layers. In contrast to some results found in the literature, our analysis reveals that the interface motion is driven by a combination of surface diffusion flux proportional to the surface Laplacian of the interface curvature and an additional contribution from nonlinear, porous-medium type bulk diffusion, For higher degenerate mobilities, bulk diffusion is subdominant. The sharp interface models are corroborated by comparing relaxation rates of perturbations to a radially symmetric stationary state with those obtained by the phase field model.

Alpha A Lee, Andreas Münch, Endre Süli

Phase field models frequently provide insight to phase transitions, and are robust numerical tools to solve free boundary problems corresponding to the motion of interfaces. A body of prior literature suggests that interface motion via surface diffusion is the long-time, sharp interface limit of microscopic phase field models such as the Cahn-Hilliard equation with a degenerate mobility function. Contrary to this conventional wisdom, we show that the long-time behaviour of degenerate Cahn-Hilliard equation with a polynomial free energy undergoes coarsening, reflecting the presence of bulk diffusion, rather than pure surface diffusion. This reveals an important limitation of phase field models that are frequently used to model surface diffusion.