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In this paper, a novel mixed spectral-Galerkin method is proposed and studied for a Maxwell transmission eigenvalue problem in a spherical domain. The method utilizes vector spherical harmonics to achieve dimension reduction. By introducing an auxiliary vector function, the original problem is rewritten as an equivalent fourth-order coupled linear eigensystem, which is further decomposed into a sequence of one-dimensional fourth-order decoupled transverse-electric (TE) and transverse-magnetic (TM) modes. Based on compact embedding theory and the spectral approximation property of compact operators, error estimates for both eigenvalue and eigenfunction approximations are established for the TE mode. For the TM mode, an efficient essential polar condition and a high-order polynomial approximation method are designed to cope with the singularity and complexity caused by the coupled boundary conditions. Numerical experiments are presented to demonstrate the efficiency and robustness of our algorithm. [ABSTRACT FROM AUTHOR]

We consider the computation of quadrature rules that are exact for a Chebyshev set of linearly independent functions on an interval [a, b]. A general theory of Chebyshev sets guarantees the existence of rules with a Gaussian property, in the sense that 2l basis functions can be integrated exactly with just l points and weights. Moreover, all weights are positive, and the points lie inside the interval [a, b]. However, the points are not the roots of an orthogonal polynomial or any other known special function as in the case of regular Gaussian quadrature. The rules are characterized by a nonlinear system of equations, and earlier numerical methods have mostly focused on finding suitable starting values for a Newton iteration to solve this system. In this paper we describe an alternative scheme that is robust and generally applicable for so-called complete Chebyshev sets. These are ordered Chebyshev sets where the first k elements also form a Chebyshev set for each k. The points of the quadrature rule are computed one by one, increasing the exactness of the rule in each step. Each step reduces to finding the unique root of a univariate and monotonic function. As such, the scheme of this paper is guaranteed to succeed. The quadrature rules are of interest for integrals with nonsmooth integrands that are not well approximated by polynomials. [ABSTRACT FROM AUTHOR]

Fourier extension is an approximation scheme in which a function on an arbitrary bounded domain is approximated using classical Fourier series on a bounding box. On the domain of interest the Fourier series exhibits redundancy, and it has the mathematical structure of a frame rather than a basis. It is not trivial to construct approximations in this frame using function evaluations in points that belong to the domain only, but one way to do so is through a discrete least squares approximation. The corresponding system is extremely ill-conditioned, due to the redundancy in the frame, yet its solution via a regularized singular value decomposition is known to be accurate to very high (and nearly spectral) precision. Still, this computation requires Q (N3) operations. In this paper we describe an algorithm to compute such Fourier extension frame approximations in only Q (N2 log2 N) operations for general 2D domains. The cost improves to Q (N log2 N) operations for simpler tensor-product domains. The algorithm exploits a phenomenon called the plunge region in the analysis of time-frequency localization operators, which manifests itself here as a sudden drop in the singular values of the least squares matrix. It is known that the size of the plunge region scales like Q (logN) in 1D problems. In this paper we show that for most 2D domains in the fully discrete case the plunge region scales like Q (N logN), proving a discrete equivalent of a result that was conjectured by Widom for a related continuous problem. The complexity estimate depends on the Minkowski or box-counting dimension of the domain boundary, and as such it is larger than Q (N logN) for domains with fractal shape. [ABSTRACT FROM AUTHOR]

Pseudo-transient continuation is a Newton-like iterative method for computing steady-state solutions of differential equations in cases where the initial data are far from a steady state. The iteration mimics a temporal integration scheme, with the time step being increased as steady state is approached. The iteration is an inexact Newton iteration in the terminal phase. In this paper we show how steady-state solutions to certain ordinary and differential algebraic equations with nonsmooth dynamics can be computed with the method of pseudo-transient continuation. An example of such a case is a discretized PDE with a Lipschitz continuous, but nondifferentiable, constitutive relation as part of the nonlinearity. In this case we can approximate a generalized derivative with a difference quotient. The existing theory for pseudo-transient continuation requires Lipschitz continuity of the Jacobian. Newton-like methods for nonsmooth equations have been globalized by trust-region methods, smooth approximations, and splitting methods in the past, but these approaches are not designed to find steady-state solutions of time-dependent problems. The method in this paper synthesizes the ideas from nonsmooth calculus and the method of pseudo-transient continuation. [ABSTRACT FROM AUTHOR]

This paper is the second part in a series on a mass conserving, high order, mixed finite element method for Stokes flow. In this part, we construct optimal hp-approximations for two classes of functions: (1) functions with general Sobolev regularity which lead to algebraic convergence rates using locally quasi-uniform meshes, and (2) singular functions belonging to countably normed (weighted) spaces which lead to exponential convergence using geometrically graded meshes. In turn, the finite element spaces we considered in part I [M. Ainsworth and C. Parker, SIAM J. Numer. Anal., 59 (2021), pp. 1218-1244] have optimal convergence properties, which is demonstrated by two numerical examples: Moffatt eddies in a wedge and flow in a T-shaped cavity. [ABSTRACT FROM AUTHOR]

Mixed finite element methods with ux errors in H(div)-norms and div-least-squares finite element methods require a separate marking strategy in obligatory adaptive mesh-refining. The refinement indicator Ψ2(T;K) = n²(T;K) + µ²(K) of a finite element domain K in an admissible triangulation T consists of some residual-based error estimator n(T;K) with some reduction property under local mesh-refining and some data approximation error µ(K). Separate marking means either Dorer marking if µ²(T) =n²(T) or otherwise an optimal data approximation algorithm with controlled accuracy. The axioms are sucient conditions on the estimators n(T;K) and data approximation errors µ(K) for optimal asymptotic convergence rates. The enfolded set of axioms of this paper simplifies [C. Carstensen, M. Feischl, M. Page, and D. Praetorius, Comput. Math. Appl., 67 (2014), pp. 1195{1253] for collective marking, treats separate marking established for the first time in an abstract framework, generalizes [C. Carstensen and E.-J. Park, SIAM J. Numer. Anal., 53 (2015), pp. 43{62] for least-squares schemes, and extends [C. Carstensen and H. Rabus, Math. Comp., 80 (2011), pp. 649{667] to the mixed finite element method with ux error control in H(div). The paper gives an outline of the mathematical analysis for optimal convergence rates but also serves as a reference so that future contributions merely verify a few axioms in a new application in order to ensure optimal mesh-refinement of the adaptive algorithm. [ABSTRACT FROM AUTHOR]

This paper studies finite element approximations of the stochastic Allen--Cahn equation with gradient-type multiplicative noise that is white in time and correlated in space. The sharp interface limit--as the diffuse interface thickness vanishes--of the stochastic Allen--Cahn equation is formally a stochastic mean curvature flow which is described by a stochastically perturbed geometric law of the deterministic mean curvature flow. Two fully discrete finite element methods which are based on different time-stepping strategies for the nonlinear term are proposed. Strong convergence with sharp rates for both fully discrete finite element methods is proved. This is done with the crucial help of the Holder continuity in time with respect to the spatial L2-norm and H¹-seminorm for the strong solution of the stochastic Allen--Cahn equation. It also relies on the fact that high moments of the strong solution are bounded in various spatial and temp oral norms. Numerical experiments are provided to gauge the performance of the proposed fully discrete finite element methods and to study the interplay of the geometric evolution and gradient-type noise. [ABSTRACT FROM AUTHOR]

The proof of optimal convergence rates of adaptive finite element methods relies on Stevenson's concept of discrete reliability. This paper proves the general discrete reliability for the nonconforming Crouzeix-Raviart finite element method on multiply connected domains in any space dimension. A novel discrete quasi-interpolation operator of first-order approximation involves an intermediate triangulation and acts as the identity on unrefined simplices, to circumvent any Helmholtz decomposition. Besides the generalization of the known application to any dimension and multiply connected domains, this paper outlines the optimality proof for uniformly convex minimization problems. This discrete reliability implies reliability for the explicit residual-based a posteriori error estimator in any space dimension and for multiply connected domains. [ABSTRACT FROM AUTHOR]

In this paper quasi-Monte Carlo (QMC) methods are applied to a class of elliptic partial differential equations (PDEs) with random coefficients, where the random coefficient is parametrized by a countably infinite number of terms in a Karhunen--Loeve expansion. Models of this kind appear frequently in numerical models of physical systems, and in uncertainty quantification. The method uses a QMC method to estimate expected values of linear functionals of the exact or approximate solution of the PDE, with the expected value considered as an infinite dimensional integral in the parameter space corresponding to the randomness induced by the random coefficient. The error analysis, arguably the first rigorous application of the modern theory of QMC in weighted spaces, exploits the regularity with respect to both the physical variables (the variables in the physical domain) and the parametric variables (the parameters corresponding to randomness). In the weighted-space theory of QMC methods, "weights," describing the varying difficulty of different subsets of the variables, are introduced in order to make sure that the high-dimensional integration problem is tractable. It turns out that the weights arising from the present analysis are of a nonstandard kind, being of neither product nor order dependent form, but instead a hybrid of the two--we refer to these as "product and order dependent weights," or "POD weights" for short. These POD weights are of a simple enough form to permit a component-by-component construction of a randomly shifted lattice rule that has optimal convergence properties for the given weighted space setting. If the terms in the expansion for the random coefficient have an appropriate decay property, and if we choose POD weights that minimize a certain upper bound on the error, then the solution of the PDE belongs to the joint function space needed for the analysis, and the QMC error (in the sense of a root-mean-square error averaged over shifts) is of order O(N-1-δ) for arbitrary δ > 0, where N denotes the number of sampling points in the parameter space. Moreover, this convergence rate and slower convergence rates under more relaxed conditions are achieved under conditions similar to those found recently by Cohen, De Vore, and Schwab [Found. Comput. Math., 10 (2010), pp. 615-646] for the same model by best N-term approximation. We analyze the impact of a finite element (FE) discretization on the overall efficiency of the scheme, in terms of accuracy versus overall cost, with results that are comparable to those of the best N-term approximation. [ABSTRACT FROM AUTHOR]

This paper establishes the equivalence of the conforming Courant finite element method, the nonconforming Crouzeix--Raviart finite element method, and several first-order discontinuous Galerkin finite element methods in the sense that the respective energy error norms are equivalent up to generic constants and higher-order data oscillations in a Poisson model problem. The Raviart--Thomas mixed finite element method is better than the previous methods, whereas the conjecture of the converse relation is proved to be false. This paper completes the analysis of comparison initiated by Braess [Calcolo, 46 (2009), pp. 149-155]. Two numerical benchmarks illustrate the comparison theorems and the possible strict superiority of the Raviart--Thomas mixed finite element method. Applications include least-squares finite element methods, finite volume methods, and equality of approximation classes for concepts of optimality for adaptive finite element methods. [ABSTRACT FROM AUTHOR]

This paper continues the convergence analysis in [M. Giles and S. Ulbrich, SIAM J. Numer. Anal., 48 (2010), pp. 882-904] of discrete approximations to the linearized and adjoint equations arising from an unsteady one-dimensional hyperbolic equation with a convex flux function. We consider a simple modified Lax-Friedrichs discretization on a uniform grid, and a key point is that the numerical smoothing increases the number of points across the nonlinear discontinuity as the grid is refined. It is proved that there is convergence in the discrete approximation of linearized output functionals even for Dirac initial perturbations and pointwise convergence almost everywhere for the solution of the adjoint discrete equations. In particular, the adjoint approximation converges to the correct uniform value in the region in which characteristics propagate into the discontinuity. Moreover, it is shown that the results of [M. Giles and S. Ulbrich, SIAM J. Numer. Anal., 48 (2010), pp. 882-904] and the present paper hold also for quite general nonlinear initial data which contain multiple shocks and for which shocks form at a later time and/or merge. [ABSTRACT FROM AUTHOR]

This paper studies a new recovery-based a posteriori error estimator for the conforming linear finite element approximation to elliptic interface problems. Instead of recovering the gradient in the continuous finite element space, the flux is recovered through a weighted L2 projection onto H(div) conforming finite element spaces. The resulting error estimator is analyzed by establishing the reliability and efficiency bounds and is supported by numerical results. This paper also proposes an adaptive finite element method based on either the recovery-based estimators or the edge estimator through local mesh refinement and establishes its convergence. In particular, it is shown that the reliability and efficiency constants as well as the convergence rate of the adaptive method are independent of the size of jumps. [ABSTRACT FROM AUTHOR]

The focus of this paper is the approximation of analytic functions on compact intervals from their pointwise values on arbitrary grids. We introduce a new method for this problem based on mapped polynomial approximation. By careful selection of the mapping parameter, we ensure both high accuracy of the approximation and an asymptotically optimal scaling of the polynomial degree with the grid spacing. As we explain, efficient implementation of this method can be achieved using nonuniform fast Fourier transforms. Numerical results demonstrate the efficiency and accuracy of this approach. [ABSTRACT FROM AUTHOR]

The present paper is a continuation of a recent article [SIAM J. Numer. Anal., 52 (2014), pp. 1867--1886], where we proposed an algorithmic approach for approximate calculation of sums of the form ΣNj=1 f(j). The method is based on a Gaussian type quadrature formula for sums, which allows the calculation of sums with a very large number of terms N to be reduced to sums with a much smaller number of summands n. In this paper we prove that the Weierstrass-Dochev- Durand--Kerner iterative numerical method, with explicitly give n initial conditions, converges to the nodes of the quadrature formula. Several methods for computin g the nodes of the discrete analogue of the Gaussian quadrature formula are compared. Since, for pr actical purposes, any approximation of a sum should use only the values of the summands f(j), we implement a simple but efficient procedure to additionally approximate the evaluations at the no des by local natural splines. Explicit numerical examples are provided. Moreover, the error in different spaces of functions is analyzed rigorously. [ABSTRACT FROM AUTHOR]

Let F be a compact set of a Banach space X. This paper analyzes the "generalized empirical interpolation method," which, given a function f ∊ F, builds an interpolant Jn[f] in an n dimensional subspace Xn ⊂ X with the knowledge of n outputs (σi(f))ni=1, where σi ∊ X' and X' is the dual space of X. The space Xn is built with a greedy algorithm that is adapted to F in the sense that it is generated by elements of F itself. The algorithm also selects the linear functionals (σi)ni=1 from a dictionary Σ ⊂ X'. In this paper, we study the interpolation error maxf∊F ‖f - Jn[f]‖X by comparing it with the best possible performance on an n dimensional space, i.e., the Kolmogorov n-width of F in X, dn(F,X). For polynomial or exponential decay rates of dn(F,X), we prove that the interpolation error has the same behavior modulo the norm of the interpolation operator. Sharper results are obtained in the case where X is a Hilbert space. [ABSTRACT FROM AUTHOR]

Low-rank approximations to large time-dependent matrices and tensors are the subject of this paper. These matrices and tensors either are given explicitly or are the unknown solutions of matrix and tensor differential equations. Based on splitting the orthogonal projection onto the tangent space of the low-rank manifold, novel time integrators for obtaining approximations by low-rank matrices and low-rank tensor trains were recently proposed. By standard theory, the Lie-Trotter and Strang projector-splitting methods are first and second order accurate, respectively, but the usual error bounds break down when the low-rank approximation has small singular values. This happens when the singular values of the solution decay without a distinct gap or when the effective rank of the solution is overestimated. On the other hand, the integrators are exact when given time-dependent matrices or tensors are already of the prescribed rank. We provide an error analysis which unifies these properties. We show that in cases where the exact solution is an ϵ-perturbation of a low-rank matrix or tensor train, the error of the projector-splitting integrator is favorably bounded in terms of ϵ and the stepsize, independently of the smallness of the singular values. Such a result does not hold for any standard integrator. Numerical experiments illustrate the theory. [ABSTRACT FROM AUTHOR]

The first-order div least squares finite element methods (LSFEMs) allow for an immediate a posteriori error control by the computable residual of the least squares functional. This paper establishes an adaptive refinement strategy based on some equivalent refinement indicators. Since the first-order div LSFEM measures the flux errors in H(div), the data resolution error measures the L² norm of the right-hand side f minus the piecewise polynomial approximation ?f without a mesh-size factor. Hence the data resolution term is neither an oscillation nor of higher order and consequently requires a particular treatment, e.g., by the thresholding second algorithm due to Binev and DeVore. The resulting novel adaptive LSFEM with separate marking converges with optimal rates relative to the notion of a nonlinear approximation class. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the numerical solution of certain matrix-valued PDEs. Such PDEs arise, for example, when constructing a Riemannian contraction metric for a dynamical system given by an autonomous ODE. We develop and analyze a new meshfree discretization scheme using kernel-based approximation spaces. However, since these approximation spaces have now to be matrix-valued, the kernels we need to use are fourth-order tensors. We will review and extend recent results on even more general reproducing kernel Hilbert spaces. We will then apply this general theory to solve a matrix-valued PDE and derive error estimates for the approximate solution. The paper ends with applications to typical examples from dynamical systems. [ABSTRACT FROM AUTHOR]

The inverse radiative transfer problem finds broad applications in medical imaging, atmospheric science, astronomy, and many other areas. This problem intends to recover optical properties, denoted as absorption and the scattering coefficient of the media, through source-measurement pairs. A typical computational approach is to form the inverse problem as a PDE-constraint optimization, with the minimizer being the to-be-recovered coefficients. The method is tested to be efficient in practice, but it lacks analytical justification: there is no guarantee of the existence or uniqueness of the minimizer, and the error is hard to quantify. In this paper, we provide a different algorithm by levering the ideas from singular decomposition analysis. Our approach is to decompose the measurements into three components, two of which encode the information of the two coefficients, respectively. We then split the optimization problem into two subproblems and use those two components to recover the absorption and scattering coefficients separately. In this regard, we prove the well-posedness of the new optimization, and the error could be quantified with better precision. In the end, we incorporate the diffusive scaling and show that the error is harder to control in the diffusive limit. [ABSTRACT FROM AUTHOR]

We develop a unfied and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate di ffusion equations ∂u-L[φ(u)] = f(x; t) in ℝN×(0; T); where L is a general symmetric Lévy-type diffusion operator. Included are both local and nonlocal problems with, e.g., L = Δ or L = -(-Δ) ff2, ff ϵ (0; 2), and porous medium, fast diffusion, and Stefan-type nonlinearities φ . By robust methods we mean that they converge even for nonsmooth solutions and under very weak assumptions on the data. We show that they are Lp-stable for p ϵ[1,∞], compact, and convergent in C([0; T];Lploc(ℝN)) for p 2 [1,∞). The first part of this project is given in [F. del Teso, J. Endal, and E. R. Jakobsen, preprint, arXiv:1801.07148v1 [math.NA], 2018] and contains the unified and easy to use theoretical framework. This paper is devoted to schemes and testing. We study many different problems and many different concrete discretizations, proving that the results of Part I apply and testing the schemes numerically. Our examples include fractional diffusions of different orders and Stefan problems, porous medium, and fast diffusion nonlinearities. Most of the convergence results and many schemes are completely new for nonlocal versions of the equation, including results on high order methods, the powers of the discrete Laplacian method, and discretizations of fast diffusions. Some of the results and schemes are new even for linear and local problems. [ABSTRACT FROM AUTHOR]

The focus of this paper is the approximation of functions which are analytic on a compact interval except at the endpoints. Typical numerical methods for approximating such functions depend upon the use of particular conformal maps from the original interval to either a semi-infinite or an infinite interval, followed by an appropriate approximation procedure on the new region. We first analyze the convergence of these existing methods and show that, in a precisely defined sense, they are suboptimal. Specifically, they exhibit poor resolution properties, by which we mean that many more degrees of freedom are required to resolve oscillatory functions than standard approximation schemes for analytic functions such as Chebyshev interpolation. To remedy this situation, we introduce two new transforms; one for each of the above settings. We provide full convergence results for these new approximations and then demonstrate that, for a particular choice of parameters, these methods lead to substantially better resolution properties. Finally, we show that optimal resolution power can be achieved by an appropriate choice of parameters, provided one forfeits classical convergence. Instead, the resulting method attains a finite, but user-controlled, accuracy specified by the parameter choice. [ABSTRACT FROM AUTHOR]

In this paper, we define new unfitted finite element methods for numerically approximating the solution of surface partial differential equations using bulk finite elements. The key idea is that the n-dimensional hypersurface, Γ ⊂ ℝn+1, is embedded in a polyhedral domain in ℝn+1 consisting of a union, Th, of (n + 1)-simplices. The unifying feature of the methodological approach is that the finite element approximating space is based on continuous piecewise linear finite element functions on the bulk triangulation Th which is independent of Γ. Our first method is a sharp interface method (SIF) which uses the bulk finite element space in an approximating weak formulation obtained from integration on a polygonal approximation, Γh, of Γ. The full gradient is used rather than the projected tangential gradient and it is this which distinguishes SIF from the method of [M. A. Olshanskii, A. Reusken, and J. Grande, SIAM J. Numer. Anal., 47 (2009), pp. 3339-3358]. The second method is a narrow band method (NBM) in which the region of integration is a narrow band of width O(h). NBM is similar to the method of [K. Deckelnick et al., IMA J. Numer. Anal., 30 (2010), pp. 351-376] but again the full gradient is used in the discrete weak formulation. The a priori error analysis in this paper shows that the methods are of optimal order in the surface L² and H¹ norms and have the advantage that the normal derivative of the discrete solution is small and converges to zero. Our third method combines bulk finite elements, discrete sharp interfaces, and narrow bands in order to give an unfitted finite element method for parabolic equations on evolving surfaces. We show that our method is conservative so that it preserves mass in the case of an advection-diffusion conservation law. Numerical results are given which illustrate the rates of convergence. [ABSTRACT FROM AUTHOR]

The explicit residual-based a posteriori error estimator for elliptic partial differential equations is reliable up to the multiplication of some generic constant which needs to be involved for full error control. The present mathematical literature takes this constant from the stability and approximation properties of Clément-type quasi-interpolation operators and so results in an overestimation of the error which is bigger than for implicit and more expensive a posterori error estimators. This paper propagates a paradigm shift to start with an equilibration error estimator technique followed by its efficiency analysis. The outcome is a refined residual-based a posteriori error estimate with explicit constants which leads to slightly sharper error control than the work of Veeser and Verfürth in 2009. A first application to guaranteed explicit error estimation for two-dimensional nonconforming and a generalization to higher-order finite element methods conclude the paper. [ABSTRACT FROM AUTHOR]

A combination of ideas in least-squares finite element methods with those of hybridized methods recently led to discontinuous Petrov--Galerkin (DPG) finite element methods. They minimize a residual inherited from a piecewise ultraweak formulation in a nonstandard, locally computable, dual norm. This paper establishes a general a posteriori error analysis for the natural norms of the DPG schemes under conditions equivalent to a priori stability estimates. It is proven that the locally computable residual norm of any discrete function is a lower and an upper error bound up to explicit data approximation errors. The presented abstract framework for a posteriori error analysis applies to known DPG discretizations of Laplace and Lamé equations and to a novel DPG method for the stress-velocity formulation of Stokes flow with symmetric stress approximations. Since the error control does not rely on the discrete equations, it applies to inexactly computed or otherwise perturbed solutions within the discrete spaces of the functional framework. Numerical illustrations show that the error control is practically feasible. [ABSTRACT FROM AUTHOR]

In this paper, we study the problem of computing the effective diffusivity for a particle moving in chaotic and stochastic flows. In addition, we numerically investigate the residual diffusion phenomenon in chaotic advection. The residual diffusion refers to the nonzero effective (homogenized) diffusion in the limit of zero molecular diffusion as a result of chaotic mixing of the streamlines. In this limit, traditional numerical methods typically fail since the solutions of the advection-diffusion equations develop sharp gradients. Instead of solving the Fokker-Planck equation in the Eulerian formulation, we compute the motion of particles in the Lagrangian formulation, which is modeled by stochastic differential equations (SDEs). We propose an effective numerical integrator based on a splitting method to solve the corresponding SDEs in which the deterministic subproblem is symplectic preserving while the random subproblem can be viewed as a perturbation. We provide rigorous error analysis for the new numerical integrator using the backward error analysis technique and show that our method outperforms standard Euler-based integrators. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several typical chaotic and stochastic flow problems of physical interest. The existence of residual diffusivity for these flow problems is also investigated. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the damped stochastic nonlinear Schrödinger (NLS) equation with multiplicative noise and its splitting-based approximation. When the damped effect is large enough, we prove that the solutions of both the damped stochastic NLS equation and the splitting scheme are exponentially stable and possess some exponential integrability. These properties show that the strong order of the scheme is 1 and independent of time. Additionally, we analyze the regularity of the Kolmogorov equation with respect to the stochastic NLS equation. As a consequence, the weak order of the scheme is shown to be 1 and independent of time. [ABSTRACT FROM AUTHOR]

We propose a new discretization method for the Stokes equations. The method is an improved version of the method recently presented in [C. Lehrenfeld and J. Schöberl, Comp. Meth. Appl. Mech. Eng., 361 (2016)] which is based on an H(div)-conforming finite element space and a hybrid discontinuous Galerkin (HDG) formulation of the viscous forces. H(div)-conformity results in favorable properties such as pointwise divergence-free solutions and pressure robustness. However, for the approximation of the velocity with a polynomial degree k, it requires unknowns of degree k on every facet of the mesh. In view of the superconvergence property of other HDG methods, where only unknowns of polynomial degree k-1 on the facets are required to obtain an accurate polynomial approximation of order k (possibly after a local postprocessing), this is suboptimal. The key idea in this paper is to slightly relax the H(div)-conformity so that only unknowns of polynomial degree k-1 are involved for normal continuity. This allows for optimality of the method also in the sense of superconvergent HDG methods. In order not to lose the benefits of H(div)-conformity, we introduce a cheap reconstruction operator which restores pressure robustness and pointwise divergence-free solutions and suits well to the finite element space with relaxed H(div)-conformity. We present this new method, carry out a thorough h-version error analysis, and demonstrate the performance of the method on numerical examples. [ABSTRACT FROM AUTHOR]

The Ensemble Kalman methodology in an inverse problems setting can be viewed as an iterative scheme, which is a weakly tamed discretization scheme for a certain stochastic differential equation (SDE). Assuming a suitable approximation result, dynamical properties of the SDE can be rigorously pulled back via the discrete scheme to the original Ensemble Kalman inversion. The results of this paper make a step towards closing the gap of the missing approximation result by proving a strong convergence result in a simplified model of a scalar SDE. We focus here on a toy model with similar properties to the one arising in the context of an Ensemble Kalman filter. The proposed model can be interpreted as a single particle filter for a linear map and thus forms the basis for further analysis. The difficulty in the analysis arises from the formally derived limiting SDE with nonglobally Lipschitz continuous nonlinearities both in the drift and in the diffusion. Here the standard Euler-Maruyama scheme might fail to provide a strongly convergent numerical scheme and taming is necessary. In contrast to the strong taming usually used, the method presented here provides a weaker form of taming. We present a strong convergence analysis by first proving convergence on a domain of high probability by using a cutoff or localization, which then leads, combined with bounds on moments for both the SDE and the numerical scheme, by a bootstrapping argument to strong convergence. [ABSTRACT FROM AUTHOR]

In this paper, we analyze standard weighted essentially nonoscillatory (WENO) reconstructions and multilevel WENO reconstructions with adaptive order (WENO-AO) using both WENO-JS and WENO-Z weighting. We also present a new WENO-AO reconstruction. We give conditions under which the reconstructions achieve optimal order accuracy for both smooth solutions and solutions with discontinuities. The old WENO-AO reconstruction drops to a fixed, base level of approximation when there are discontinuities in the solution, but the new one maintains the accuracy of the largest stencil over which the solution is smooth. Our analysis in the discontinuous case requires that the smoothness indicators do not approach zero as the grid is refined. We provide a condition to ensure this result, but we also show an example where this can fail to occur. That is, we show that WENO reconstructions can fail to maintain the order of approximation of the smallest stencil over which the solution is smooth. We also present numerical results confirming the convergence theory of the old and new WENO-AO reconstructions and compare their performance in solving conservation laws. [ABSTRACT FROM AUTHOR]

Dynamical low-rank approximation in the Tucker tensor format of given large timedependent tensors and of tensor differential equations is the subject of this paper. In particular, a discrete time integration method for rank-constrained Tucker tensors is presented and analyzed. It extends the known projector-splitting integrator for dynamical low-rank approximation of matrices to Tucker tensors and is shown to inherit the same favorable properties. The integrator is based on iteratively applying the matrix projector-splitting integrator to tensor unfoldings but with inexact solution in a substep. It has the property that it reconstructs time-dependent Tucker tensors of the given rank exactly. The integrator is also shown to be robust to the presence of small singular values in the tensor unfoldings. Numerical examples with time-dependent problems from quantum physics and tensor optimization methods illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a numerical solution of partial differential equations on the unit sphere is given by using a kernel trial approximation in combination with a special Petrov-Galerkin test discretization. The solvability of the scheme is proved, and the error bounds are obtained for functions in appropriate Sobolev spaces. The condition number of the final system is estimated in terms of discretization parameters. The method is meshless because in the trial side the numerical solution parameterizes entirely in terms of scattered points and in the test side everything breaks down to simple numerical integrations over independent spherical caps. This means that no connected background mesh is required for either approximation or integration. [ABSTRACT FROM AUTHOR]

This paper investigates a first-order and a second-order approximation technique for the shallow water equation with topography using continuous finite elements. Both methods are explicit in time and are shown to be well-balanced. The first-order method is invariant domain preserving and satisfies local entropy inequalities when the bottom is flat. Both methods are positivity preserving. Both techniques are parameter free, work well in the presence of dry states, and can be made high order in time by using strong stability preserving time stepping algorithms. [ABSTRACT FROM AUTHOR]

In numerous applications, scientists and engineers acquire varied forms of data that partially characterize the inputs to an underlying physical system. This data is then used to inform decisions such as controls and designs. Consequently, it is critical that the resulting control or design is robust to the inherent uncertainties associated with the unknown probabilistic characterization of the model inputs. In this paper, we consider optimal control and design problems constrained by partial differential equations with uncertain inputs. We do not assume a known probabilistic model for the inputs, but rather we formulate the problem as a distributionally robust optimization problem where the outer minimization problem determines the control or design, while the inner maximization problem determines the worst-case probability measure that matches desired characteristics of the data. We analyze the inner maximization problem in the space of measures and introduce a novel measure approximation technique, based on the approximation of continuous functions, to discretize the unknown probability measure. We prove consistency of our approximated min-max problem and conclude with numerical results. [ABSTRACT FROM AUTHOR]

The traditional wave equation models wave propagation in an ideal medium. To characterize wave propagation in inhomogeneous media with frequency-dependent power-law attenuation, the space-time fractional wave equation is needed; further incorporating the additive white Gaussian noise (external noise) coming from many natural sources leads to the stochastic space-time fractional wave equation. This paper discusses the Galerkin finite element approximations for the stochastic space-time fractional wave equation forced by an additive space-time white noise. We first discretize the space-time additive noise, which introduces a modeling error and results in a regularized stochastic space-time fractional wave equation; then the regularity of the regularized equation is analyzed. For the discretization in space, the finite element approximation is used and the definition of the discrete fractional Laplacian is introduced. We derive the mean-squared L²-norm a priori estimates for the modeling error and for the approximation error to the solution of the regularized problem; numerical experiments are then performed to confirm the estimates. For the time-stepping, we calculate the analytical solution of the space semidiscretized equation, which is expressed by Mittag-Lever functions. [ABSTRACT FROM AUTHOR]

This paper focuses on the numerical analysis for three-dimensional Bean's critical-state model in type-II superconductivity. We derive hyperbolic mixed variational inequalities of the second kind for the evolution Maxwell equations with Bean's constitutive law between the electric Field and the current density. On the basis of the variational inequality in the magnetic induction formulation, a semidiscrete Ritz-Galerkin approximation problem is rigorously analyzed, and a strong convergence result is proven. Thereafter, we propose a concrete realization of the Ritz-Galerkin approximation through a mixed Finite element method based on edge elements of Nédélec's First family, Raviart-Thomas face elements, divergence-free Raviart-Thomas face elements, and piecewise constant elements. As a Final result, we prove error estimates for the proposed mixed Finite element method. [ABSTRACT FROM AUTHOR]

This paper derives a posteriori error estimates for conforming numerical approximations of the Laplace eigenvalue problem with a homogeneous Dirichlet boundary condition. In particular, upper and lower bounds for an arbitrary simple eigenvalue are given. These bounds are guaranteed, fully computable, and converge with optimal speed to the given exact eigenvalue. They are valid without restrictions on the computational mesh or on the approximate eigenvector; we only need to assume that the approximate eigenvalue is separated from the surrounding smaller and larger exact ones, which can be checked in practice. Guaranteed, fully computable, optimally convergent, and polynomial-degree robust bounds on the energy error in the approximation of the associated eigenvector are derived as well, under the same hyp otheses. Remarkably, there appears no unknown (solution-, regularity-, or polynomial-degree-dependent) constant in our theory, and no convexity/regularity assumption on the computational domain/exact eigenvector(s) is needed. The multiplicative constant appearing in our estimates depends on (computable estimates of) the gaps to the surrounding exact eigenvalues. Its two improvements are presented. First, it is reduced by a fixed factor under an explicit, a posteriori calculable condition on the mesh and on the approximate eigenvector--eigenvalue pair. Second, when an elliptic regularity assumption on the corresponding source problem is satisfied with known constants, this multiplicative constant can be brought to the optimal value of one. Inexact algebraic solvers are taken into account; the estimates are valid on each iteration and can serve for the design of adaptive stopping criter ia. The application of our framework to conforming finite element approximations of arbitrary polynomial degree is provided, along with a numerical illustration on a set of test problems. [ABSTRACT FROM AUTHOR]

Monotonicity-preserving approximation methods are used in numerous applications. With them we can reconstruct a function from a discrete set of data while preserving its monotonicity properties. In this paper we analyze monotone piecewise cubic Hermite interpolants for uniform and nonuniform grids. We present and analyze two new methods to prescribe the derivatives at the breakpoints that lead to third order approximations even in the case of nonuniform grids. We demonstrate that with these alternatives the monotonicity may only be lost in rather extreme situations. In such cases we propose modifications of the algorithms that guarantee the monotonicity but with local second order accuracy. We also perform several numerical experiments which exemplify the properties of the proposed algorithms and compare them with the technique proposed by Fritsch and Butland, in which the derivatives at the breakpoints are calculated using Brodlie's function. It is available in Netlib (PCHIP.FOR) and it is also used in the PCHIP program of MATLAB [C. Moler, Numerical Computing with MATLAB, SIAM, Philadelphia, 2004; G. Wolberg and I. Alfy, J. Comput. Appl. Math., 143 (2002), pp. 145-188]. With this last approach, third order approximations to smooth functions are obtained for uniform grids, but when the grid is nonuniform, only second order approximations are attained. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical approximation of the minimizer of the continuous Rudin--Osher--Fatemi (ROF) model for image denoising. A new discrete total variation is proposed and the associated Hilbertian total variation denoising model is used to construct continuous piecewise linear functions that approximate the minimizer of the ROF model in the strong topology of L2(Ω), provided that the data function is bounded and weakly regular in the sense of Lip(α, L² (Ω)). [ABSTRACT FROM AUTHOR]

The present paper is concerned with the unilateral contact model in linear elastostatics, the so-called Signorini problem. (Our results can also be applied to the scalar Signorini problem.) A standard continuous linear finite element approximation is first chosen to approach the two-dimensional problem. We develop a new error analysis in the H¹-norm using estimates on Poincaré constants with respect to the size of the areas of the noncontact sets. In particular we do not assume any additional hypothesis on the finiteness of the set of transition points between contact and noncontact. This approach allows us to establish better error bounds under sole Hτ assumptions on the solution: if 3/2 < τ < 2 we improve the existing rate by a factor h(τ-3/2)² and if τ = 2 the existing rate (h¾) is improved by a new rate of h √"ln(h)|. Using the same finite element spaces as previously we then consider another discrete approximation of the (nonlinear) contact condition in which the same kind of analysis leads to the same convergence rates as for the first approximation. [ABSTRACT FROM AUTHOR]

This paper presents mixed finite element methods of higher-order for a simplified Signorini problem and an idealized frictional problem. The discretization is based on a mixed variational formulation proposed by Haslinger et al., which is extended to higher-order finite elements. To guarantee the unique existence of the solution of the mixed method, a discrete inf-sup condition is proven. Approximation results of the p-method of finite elements and some inverse estimates for higher-order polynomials are applied. Numerical results confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

This paper is devoted to the numerical approximation of Lyapunov and Sacker-Sell Spectral intervals for linear differential-algebraic equations (DAEs). The spectral analysis for DAEs is improved and the concepts of leading directions and solution subspaces associated with spectral intervals are extended to DAEs. Numerical methods based on smooth singular value decompositions are introduced for Computing all or only some spectral intervals and their associated leading directions. The numerical algorithms as well as implementation issues are discussed in detail and Numerical examples are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the possibility of approximating the stationary solution of a stochastic differential equation (SDE). We start with the random dynamical system generated by the SDE with the multiplicative noise. We prove that the pullback flow has a stationary point. However, the stationary point is not constructible explicitly; therefore, we look at the numerical approximation. We prove that the discrete time random dynamical system also has a stationary point. Finally, we prove mean-square convergence of the approximate stationary solution to the exact stationary solution as the time step diminishes, as well as almost surely convergence when the time step is rational. [ABSTRACT FROM AUTHOR]

Liu, Osher, and Chan introduced weighted essentially nonoscillatory (WENO) reconstructions in [X.-D. Liu, S. Osher, and T. Chan, J. Comput. Phys., 115 (1994), pp. 200-212] to improve the order of accuracy of essentially nonoscillatory (ENO) reconstructions [A. Harten et al., J. Comput. Phys., 71 (1987), pp. 231-303]. In [G.-S. Jiang and C.-W. Shu, J. Comput. Phys., 126 (1996), pp. 202-228], the authors proposed smoothness indicators to obtain a WENO fifth order reconstruction from third order ENO reconstructions. With these smoothness indicators, Balsara and Shu [J. Comput. Phys., 160 (2000), pp. 405-452] and, later, [G. A. Gerolymos, D. Sénéchal, and I. Vallet, J. Comput. Phys., 228 (2009), pp. 8481-8524] obtained (2r - 1)th order WENO recoilstructions from rth order ENO reconstructions for 4 < r < 6, resp., 7 < r < 9. In [A. K. Henrick, T. D. Aslam, and J. M. Powers, J. Comput. Phys., 207 (2005), pp. 542-567], the authors noticed that these reconstructions do not attain the optimal order 2r - 1 at extrema and they proposed a fix for the problem. Other authors [R. Borges et al., J. Comput. Phys., 227 (2008), pp. 3191-3211; N. K. Yamaleev and M. H. Carpenter, d. Comput. Phys., 228 (2009), pp. 4248-4272] have addressed this problem with different weight designs for Jiang-Shu smoothness indicators. In this paper we exploit the special structure of Jiang-Shu smoothness indicators and analyze the role of a parameter appearing in the weight definition to avoid division by zero to obtain for any r > 2, by standard approximation properties of Lagrange interpolation, that the order of the WENO reconstruction is 2r - 1 at smooth regions, regardless of neighboring extrema, whilst this order is r, as ENO reconstructions have, when the function has a discontinuity in the stencil of 2r - 1 points but it is smooth in at least one of the substencils of r points. The optimal weights are also obtained in closed form. [ABSTRACT FROM AUTHOR]

We study a block triangular preconditioner for finite element approximations of the linearized Navier-Stokes equations. The preconditioner is based on the augmented Lagrangian formulation of the problem and was introduced by the authors in [SIAM J. Sci. Comput., 28 (2006), pp. 2095-2113]. In this paper we prove field-of-values type estimates for the preconditioned system which lead to optimal convergence bounds for the GMRES algorithm applied to solve the system. Two variants of the preconditioner are considered: an ideal one based on exact solves for the velocity submatrix, and a more practical variant based on block triangular approximations of the velocity submatrix. [ABSTRACT FROM AUTHOR]

A theoretical analysis of a first-order div least-squares finite element method is presented. Our main interest is providing L2-norm error estimates for the primary function u and the flux σ = -효∇u. While there has been significant progress in the error estimates for the flux in L2-norm, these estimates have drawbacks, such as requiring smooth solutions or that the error estimate for ||σ - σh||0 is coupled with the error estimate for ||u - uh||1, where (uh, σh) is the least squares approximate solution for (u, σ = - 효 ∇u). In this paper, with the minimum regularity assumption u ∈ H1+α, α > 1/2, we separate the error estimate for ||σ - ||h||0 from ||u - uh||1 and establish that ||σ - σh||0 is bounded by the best approximation for a in the finite element space and ||u - uhh||0. Then, we proceed to obtain two new estimates for ||u - uh||0. The first estimate is useful to understand how the accuracy of ||u - uh||0 affects the accuracy of ||σ - σh||0 and the other sheds new light on the convergence behavior of ||u - uh||0. Our analysis does not require smooth solutions and the domain is allowed to be nonconvex. The resulting error estimates are valid for both smooth and low regularity solutions and show that optimal rates of convergence can be achieved by choosing proper finite element spaces such as the Raviart Thomas spaces for the flux σ. [ABSTRACT FROM AUTHOR]

The coupled problem with Stokes flow in one subdomain and a Darcy flow model in a second subdomain is studied in this paper. Both flow problems are treated as first-order systems, involving pseudostress and velocity in the Stokes case and using a flux-pressure formulation in the Darcy subdomain as process variables, respectively. The Beavers-Joseph-Saffman interface conditions are treated by an appropriate interface functional which is added to the least squares functional associated with the subdomain problems. A combination of H(div)-conforming Raviart Thomas and standard H¹-conforming elements is used for the Stokes as well as for the Darcy subsystem. The homogeneous least squares functional is shown to be equivalent to an appropriate norm allowing the use of standard finite element approximation estimates. It also establishes the fact that the local evaluation of the least squares functional itself constitutes an a posteriori error estimator to be used for adaptive refinement strategies. [ABSTRACT FROM AUTHOR]

This paper proposes and analyzes some fully discrete (multiphysics) finite element methods for a displacement-pressure model which describes swelling dynamics of polymer gels under mechanical constraints. By introducing an "elastic pressure" we first present a reformulation of the original model. We then propose a time-stepping scheme which decouples the PDE system at each time step into two subproblems, one of which is a Stokes-like problem for the displacement vector field (of the solid network of the gel) and the other is a diffusion problem for a "pseudopressure" field (of the solvent of the gel). To make such a multiphysics approach feasible, it is vital to find admissible constraints to resolve the uniqueness issue for the Stokes-like problem and to construct a good boundary condition for the diffusion equation so that it also becomes uniquely solvable. The key to the first difficulty is to discover certain conserved quantities for the PDE solution, and the solution to the second difficulty is to use the Stokes-like problem to generate a boundary condition for the diffusion problem. The time-stepping scheme allows one to use any convergent Stokes solver (and its code) together with any convergent diffusion equation solver (and its code) to solve the polymer gel model. In the paper, the Taylor-Hood mixed finite element method combined with the continuous linear finite element method are chosen as an example to present the ideas and to demonstrate the viability of the proposed multiphysics approach. It is proved that, under a mesh constraint, both the semidiscrete (in space) and fully discrete methods enjoy some discrete energy laws which mimic the differential energy law satisfied by the PDE solution. Optimal order error estimates in various norms are established for both semidiscrete and fully discrete methods. Numerical experiments are also presented to show the performance of the proposed approach and methods. [ABSTRACT FROM AUTHOR]

This paper analyzes the convergence of discrete approximations to the linearized equations arising from an unsteady one-dimensional hyperbolic equation with a convex flux function. A simple modified Lax-Friedrichs discretization is used on a uniform grid, and a key point is that the numerical smoothing increases the number of points across the nonlinear discontinuity as the grid is refined. It is proved that this gives pointwise convergence almost everywhere for the solution to the linearized discrete equations with smooth initial data, and also convergence in the discrete approximation of linearized output functionals. In Part 2 [M. Giles and S. Ulbrich, SIAM J. Numer. Anal., 48 (2010), pp. 905-921] we extend the results to Dirac initial data for the linearized equation and will prove the pointwise convergence almost everywhere for the solution of the adjoint discrete equations. In addition, we present numerical results illustrating the asymptotic behavior which is analyzed. [ABSTRACT FROM AUTHOR]

This paper takes another look at the convergence analysis of the Arnoldi procedure for solving non-Hermitian eigenvalue problems. Two main viewpoints are put in contrast. The first exploits the eigenbasis, when there is one, and relies on classical min-max approximation theory results. The second approach relies on the Schur factorization. Its aim is to link the convergence analysis of the Arnoldi process for eigenvalue problems to that of the generalized minimal residual iterations (GMRES), for which much is known. [ABSTRACT FROM AUTHOR]