Showing papers in "Ima Journal of Numerical Analysis in 2016"

Conforming and nonconforming virtual element methods for elliptic problems

Abstract: EPSRC (Grant EP/L022745/1) to A.C.; Laboratory Directed Research and Development program (LDRD), US Department of Energy Office of Science, Office of Fusion Energy Sciences, under the auspices of the National Nuclear Security Administration of the US Department of Energy by Los Alamos National Laboratory, operated by Los Alamos National Security LLC under contract DE-AC52- 06NA25396 to G.M.; Ph.D. Studentship from the College of Science and Engineering at the University of Leicester and an EPSRC Doctoral Training Grant to O.S.

Stability and convergence of a second-order mixed finite element method for the Cahn–Hilliard equation

Abstract: In this paper we devise and analyze an unconditionally stable, second-order-in-time numerical scheme for the Cahn-Hilliard equation in two and three space dimensions. We prove that our two-step scheme is unconditionally energy stable and unconditionally uniquely solvable. Furthermore, we show that the discrete phase variable is bounded in $L^\infty (0,T;L^\infty)$ and the discrete chemical potential is bounded in $L^\infty (0,T;L^2)$, for any time and space step sizes, in two and three dimensions, and for any finite final time $T$. We subsequently prove that these variables converge with optimal rates in the appropriate energy norms in both two and three dimensions. We include in this work a detailed analysis of the initialization of the two-step scheme.

Weak Galerkin method for the coupled Darcy–Stokes flow

Abstract: A family of weak Galerkin finite element discretization is developed for solving the coupled Darcy-Stokes equation. The equation in consideration admits the Beaver-Joseph-Saffman condition on the interface. By using the weak Galerkin approach, in the discrete space we are able to impose the normal continuity of velocity explicitly. Or in other words, strong coupling is achieved in the discrete space. Different choices of weak Galerkin finite element spaces are discussed, and error estimates are given.

A superconvergent HDG method for the Incompressible Navier-Stokes Equations on general polyhedral meshes

Abstract: We present a superconvergent hybridizable discontinuous Galerkin (HDG) method for the steady-state incompressible Navier-Stokes equations on general polyhedral meshes. For arbitrary conforming polyhedral mesh, we use polynomials of degree k+1, k, k to approximate the velocity, velocity gradient and pressure, respectively. In contrast, we only use polynomials of degree k to approximate the numerical trace of the velocity on the interfaces. Since the numerical trace of the velocity field is the only globally coupled unknown, this scheme allows a very efficient implementation of the method. For the stationary case, and under the usual smallness condition for the source term, we prove that the method is well defined and that the global L2-norm of the error in each of the above-mentioned variables and the discrete H1-norm of the error in the velocity converge with the order of k+1 for k>=0. We also show that for k>=1, the global L2-norm of the error in velocity converges with the order of k+2. From the point of view of degrees of freedom of the globally coupled unknown: numerical trace, this method achieves optimal convergence for all the above-mentioned variables in L2-norm for k>=0, superconvergence for the velocity in the discrete H1-norm without postprocessing for k>=0, and superconvergence for the velocity in L2-norm without postprocessing for k>=1.

Gradient discretization of hybrid-dimensional Darcy flow in fractured porous media with discontinuous pressures at matrix-fracture interfaces

Abstract: We investigate the discretization of Darcy flow through fractured porous media on general meshes. We consider a hybrid dimensional model, invoking a complex network of planar fractures. The model accounts for matrix-fracture interactions and fractures acting either as drains or as barriers, i.e. we have to deal with pressure discontinuities at matrix-fracture interfaces. The numerical analysis is performed in the general framework of gradient discretizations which is extended to the model under consideration. Two families of schemes namely the Vertex Approximate Gradient scheme (VAG) and the Hybrid Finite Volume scheme (HFV) are detailed and shown to satisfy the gradient scheme framework, which yields, in particular, convergence. Numerical tests confirm the theoretical results. Gradient Discretization; Darcy Flow, Discrete Fracture Networks, Finite Volume

H(div) conforming and DG methods for incompressible Euler’s equations

Abstract: H(div) conforming and discontinuous Galerkin (DG) methods are designed for incompressible Euler’s equation in two and three dimension. Error estimates are proved for both the semi-discrete method and fully-discrete method using backward Euler time stepping. Numerical examples exhibiting the performance of the methods are given.

Nonsmooth trust region algorithms for locally Lipschitz functions on Riemannian manifolds

Abstract: This paper presents a Riemannian trust region algorithm for unconstrained optimization problems with locally Lipschitz objective functions defined on complete Riemannian manifolds. To this end we define a function $\Phi :TM\rightarrow \mathbb {R}$ on the tangent bundle $TM$ , and at the $k$ th iteration, using the restricted function $\Phi |_{T_{x_k}M}$ , where $T_{x_k}M$ is the tangent space at $x_k$ , a local model function $Q_k$ that carries both first- and second-order information for the locally Lipschitz objective function $f:M\rightarrow \mathbb {R}$ on a Riemannian manifold $M$ , is defined and minimized over a trust region. We establish the global convergence of the proposed algorithm. Moreover, using the Riemannian $\varepsilon $ -subdifferential, a suitable model function is defined. Numerical experiments illustrate our results.

On approximations of the curve shortening flow and of the mean curvature flow based on the DeTurck trick

Abstract: In this paper we discuss novel numerical schemes for the computation of the curve shortening and mean curvature flows that are based on special reparametrizations. The main idea is to use special solutions to the harmonic map heat flow in order to reparametrize the equations of motion. This idea is widely known from the Ricci flow as the DeTurck trick. By introducing a variable time scale for the harmonic map heat flow, we obtain families of numerical schemes for the reparametrized flows. For the curve shortening flow this family unveils a surprising geometric connection between the numerical schemes in [5] and [9]. For the mean curvature flow we obtain families of schemes with good mesh properties similar to those in [3]. We prove error estimates for the semi-discrete scheme of the curve shortening flow. The behaviour of the fully-discrete schemes with respect to the redistribution of mesh points is studied in numerical experiments. We also discuss possible generalizations of our ideas to other extrinsic flows.

A penalty-free Nitsche method for the weak imposition of boundary conditions in compressible and incompressible elasticity

Abstract: In this paper, we study the stability of the nonsymmetric version of the Nitsche method without penalty for compressible and incompressible elasticity. For the compressible case we prove convergence of the error in the H1- and L2-norms. In the incompressible case we use a Galerkin least squares pressure stabilization and we prove convergence in the H1-norm for the velocity and convergence of the pressure in the L2-norm.

Numerical Approximation of Fractional Powers of Regularly Accretive Operators

Abstract: We study the numerical approximation of fractional powers of accretive operators in this paper. Namely, if A is the accretive operator asso- ciated with an accretive sesquilinear form A(·,·) defined on a Hilbert space V contained in L 2 (), we approximate A −� for � 2 (0,1). The fractional powers are defined in terms of the so-called Balakrishnan integral formula. Given a finite element approximation space VhV, A −� is approximated by A −� hh where Ah is the operator associated with the form A(·,·) restricted to Vh andh is the L 2 ()-projection onto Vh. We first provide error esti- mates for (AA � �h)f in Sobolev norms with index in (0,1) for appropriate f. These results depend on elliptic regularity properties of variational solu- tions involving the form A(·,·) and are valid for the case of less than full elliptic regularity. We also construct and analyze an exponentially convergent SINC quadrature approximation to the Balakrishnan integral defining A � �hf. Fi- nally, the results of numerical computations illustrating the proposed method are given.

Error estimates for a numerical approximation to the compressible barotropic Navier-Stokes equations

Abstract: We present here a general method based on the investigation of the relative energy of the system, that provides an unconditional error estimate for the approximate solution of the barotropic Navier Stokes equations obtained by time and space discretization. We use this methodology to derive an error estimate for a specific DG/finite element scheme for which the convergence was proved in [26].

An unconditionally monotone numerical scheme for the two-factor uncertain volatility model

Abstract: Under the assumption that two asset prices follow an uncertain volatility model, the maximal and 2 minimal solution values of an option contract are given by a two dimensional Hamilton-Jacobi-Bellman 3 (HJB) PDE. A fully implicit, unconditionally monotone nite dierence numerical scheme is developed 4 in this paper. Consequently, there are no time step restrictions due to stability considerations. The 5 discretized algebraic equations are solved using policy iteration. Our discretization method results in a 6 local objective function which is a discontinuous function of the control. Hence some care must be taken 7 when applying policy iteration. The main diculty in designing a discretization scheme is development 8 of a monotonicity preserving approximation of the cross derivative term in the PDE. We derive a hybrid 9 numerical scheme which combines use of a xed point stencil and a wide stencil based on a local coordinate 10 rotation. The algorithm uses the xed point stencil as much as possible to take advantage of its accuracy 11 and computational eciency. The analysis shows that our numerical scheme is l1 stable, consistent in 12 the viscosity sense, and monotone. Thus, our numerical scheme guarantees convergence to the viscosity 13 solution. 14

Approximation of the invariant law of SPDEs: error analysis using a Poisson equation for a full-discretization scheme

Abstract: We study the long-time behavior of fully discretized semilinear SPDEs with additive space-time white noise, which admit a unique invariant probability measure $\mu$. We show that the average of regular enough test functions with respect to the (possibly non unique) invariant laws of the approximations are close to the corresponding quantity for $\mu$. More precisely, we analyze the rate of the convergence with respect to the different discretization parameters. Here we focus on the discretization in time thanks to a scheme of Euler type, and on a Finite Element discretization in space. The results rely on the use of a Poisson equation; we obtain that the rates of convergence for the invariant laws are given by the weak order of the discretization on finite time intervals: order $1/2$ with respect to the time-step and order $1$ with respect to the mesh-size.

A priori error analysis of space–time Trefftz discontinuous Galerkin methods for wave problems

Abstract: We present and analyse a space–time discontinuous Galerkin method for wave propagation problems. The special feature of the scheme is that it is a Trefftz method, namely that trial and test functions are solution of the partial differential equation to be discretised in each element of the (space–time) mesh. The method considered is a modification of the discontinuous Galerkin schemes of Kretzschmar et al. (2014) and of Monk & Richter (2005). For Maxwell’s equations in one space dimension, we prove stability of the method, quasi-optimality, best approximation estimates for polynomial Trefftz spaces and (fully explicit) error bounds with high order in the meshwidth and in the polynomial degree. The analysis framework also applies to scalar wave problems and Maxwell’s equations in higher space dimensions. Some numerical experiments demonstrate the theoretical results proved and the faster convergence compared to the non-Trefftz version of the scheme.

A high-order time-parallel scheme for solving wave propagation problems via the direct construction of an approximate time-evolution operator

Abstract: Our manuscript demonstrates a technique for efficiently solving the classical wave equation, the shallow water equations, and, more generally, equations of the form ∂u/∂t=Lu∂u/∂t=Lu, where LL is a skew-Hermitian differential operator. The idea is to explicitly construct an approximation to the time-evolution operator exp(τL)exp(τL) for a relatively large time-step ττ. Recently developed techniques for approximating oscillatory scalar functions by rational functions, and accelerated algorithms for computing functions of discretized differential operators are exploited. Principal advantages of the proposed method include: stability even for large time-steps, the possibility to parallelize in time over many characteristic wavelengths and large speed-ups over existing methods in situations where simulation over long times are required. Numerical examples involving the 2D rotating shallow water equations and the 2D wave equation in an inhomogenous medium are presented, and the method is compared to the 4th order Runge–Kutta (RK4) method and to the use of Chebyshev polynomials. The new method achieved high accuracy over long-time intervals, and with speeds that are orders of magnitude faster than both RK4 and the use of Chebyshev polynomials.

Lebesgue constants arising in a class of collocation methods

Abstract: Estimates are obtained for the Lebesgue constants associated with the Gauss quadra- ture points on (i1, +1) augmented by the point i1 and with the Radau quadrature points on either (i1, +1) or (i1, +1). It is shown that the Lebesgue constants are O( p N), where N is the number of quadrature points. These point sets arise in the estimation of the residual associated with recently developed orthogonal collocation schemes for optimal control problems. For problems with smooth solutions, the estimates for the Lebesgue constants can imply an exponential decay of the residual in the collocated problem as a function of the number of quadrature points. tablished for these schemes. The analysis is based on a bound for the inverse of a linearized operator associated with the discretized problem, and an estimate for the residual one gets when substituting the solution to the continuous problem into the discretized problem. This paper focuses on the estimation of the residual. We show that the residual in the sup-norm is bounded by the sup-norm distance between the derivative of the solution to the continuous problem and the derivative of the inter- polant of the solution. By Markov's inequality (18), this distance can be bounded in terms of the Lebesgue constant for the point set and the error in best polynomial approximation. A classic result of Jackson (17) gives an estimate for the error in best approximation. The Lebesgue constant that we need to analyze corresponds to the roots of a Jacobi polynomial on (−1, +1) augmented by either ? = +1 or ? = −1. The effects of the added endpoints were analyzed by Vertesi in (24). For either the Gauss quadrature points on (−1, +1) augmented by ? = +1 or the Radau quadrature points on (−1, +1) or on (−1, +1), the bound given in (24, Thm. 2.1) for the Lebesgue constants is O(log(N) √ N), where N is the number of quadrature points. We sharpen this bound to O( √ N). To motivate the relevance of the Lebesgue constant to collocation methods, let us consider the scalar first-order differential equation u

Gradient Schemes for Stokes problem

Abstract: We provide a framework which encompasses a large family of conforming and nonconforming numerical schemes, for the approximation of the steady state incompressible Stokes equations with homogeneous Dirichlet’s boundary conditions. Three examples (Taylor-Hood, extended MAC and Crouzeix-Raviart schemes) are shown to enter into this framework. The convergence of the scheme is proved by compactness arguments, thanks to estimates on the discrete solution that allow to prove the weak convergence to the unique continuous solution of the problem. Then strong convergence results are obtained thanks to the limit problem. An error estimate result is provided, applying on solutions with low regularity.

Finite volume element method for two-dimensional fractional subdiffusion problems

Abstract: In this paper, a semi-discrete spatial finite volume (FV) method is proposed and analyzed for approximating solutions of anomalous subdiffusion equations involving a temporal fractional derivative of order $\alpha \in (0,1)$ in a two-dimensional convex polygonal domain. Optimal error estimates in $L^\infty(L^2)$- norm is shown to hold. Superconvergence result is proved and as a consequence, it is established that quasi-optimal order of convergence in $L^{\infty}(L^{\infty})$ holds. We also consider a fully discrete scheme that employs FV method in space, and a piecewise linear discontinuous Galerkin method to discretize in temporal direction. It is, further, shown that convergence rate is of order $O(h^2+k^{1+\alpha}),$ where $h$ denotes the space discretizing parameter and $k$ represents the temporal discretizing parameter. Numerical experiments indicate optimal convergence rates in both time and space, and also illustrate that the imposed regularity assumptions are pessimistic.

Hybrid expansion–contraction: a robust scaleable method for approximating the H∞ norm

Abstract: We present a new scaleable algorithm for approximating the $H_{\infty }$ norm, an important robust stability measure for linear dynamical systems with input and output. Our spectral-value-set-based method uses a novel hybrid expansion–contraction scheme that, under reasonable assumptions, is guaranteed to converge to a stationary point of the optimization problem defining the $H_{\infty }$ norm, and, in practice, typically returns local or global maximizers. We prove that the hybrid expansion–contraction method has a quadratic rate of convergence that is also confirmed in practice. In comprehensive numerical experiments, we show that our new method is not only robust but exceptionally fast, successfully completing a large-scale test set 25 times faster than an earlier method by Guglielmi, Gurbuzbalaban & Overton (2013, SIAM J. Matrix Anal. Appl., 34, 709–737), which occasionally breaks down far from a stationary point of the underlying optimization problem.

Spectral discretization of Darcy's equations coupled with the heat equation

Abstract: In this paper we consider the heat equation coupled with Darcy's law with a nonlin-ear source term describing heat production due to an exothermic chemical reaction. Existence and uniqueness of a solution are established. Next, a spectral discretization of the problem is presented and thoroughly analysed. Finally, we present some numerical experiments which confirm the interest of the discretization.

High-order accurate, fully discrete entropy stable schemes for scalar conservation laws

Abstract: The recently developed TECNO schemes for hyperbolic conservation laws are designed to be high-order accurate and entropy stable, but are, as of yet, only semi-discrete. We perform an explicit discretization of the temporal derivative to obtain a fully discrete scheme, and derive a non-strict CFL condition that ensures global entropy stability. The scheme is tested in a series of numerical experiments.

A $C^0$ linear finite element method for two fourth-order eigenvalue problems

Abstract: In this article, we construct a C0 linear finite element method for two fourth-order eigenvalue problems: the biharmonic and the transmission eigenvalue problems. The basic idea of our construction is to use gradient recovery operator to compute the higher-order derivatives of a C0 piecewise linear function, which do not exist in the classical sense. For the biharmonic eigenvalue problem, the optimal convergence rates of eigenvalue/eigenfunction approximation are theoretically derived and numerically verified. For the transmission eigenvalue problem, the optimal convergence rate of the eigenvalues is verified by two numerical examples: one for constant refraction index and the other for variable refraction index. Compared with existing schemes in the literature, the proposed scheme is straightforward and simpler, and computationally less expensive to achieve the same order of accuracy.