Showing papers in "SIAM Journal on Numerical Analysis in 2015"
TL;DR: These schemes, based on stabilization and convex splitting, are the first, to the best of the authors' knowledge, totally decoupled, linear, unconditionally energy stable schemes for phase-field models of two-phase incompressible flows.
Abstract: In this paper we construct two classes, based on stabilization and convex splitting, of decoupled, unconditionally energy stable schemes for Cahn--Hilliard phase-field models of two-phase incompressible flows. At each time step, these schemes require solving only a sequence of elliptic equations, including a pressure Poisson equation. Furthermore, all of these elliptic equations are linear for the schemes based on stabilization, making them the first, to the best of the authors' knowledge, totally decoupled, linear, unconditionally energy stable schemes for phase-field models of two-phase incompressible flows. Thus, the schemes constructed in this paper are very efficient and easy to implement.
210 citations
TL;DR: In this article, the authors presented new immersed finite element (IFE) methods for solving the popular second order elliptic interface problems on structured Cartesian meshes even if the involved interfaces have nontrivial geometries.
Abstract: This article presents new immersed finite element (IFE) methods for solving the popular second order elliptic interface problems on structured Cartesian meshes even if the involved interfaces have nontrivial geometries. These IFE methods contain extra stabilization terms introduced only at interface edges for penalizing the discontinuity in IFE functions. With the enhanced stability due to the added penalty, not only can these IFE methods be proven to have the optimal convergence rate in an energy norm provided that the exact solution has sufficient regularity, but also numerical results indicate that their convergence rates in both the $H^1$-norm and the $L^2$-norm do not deteriorate when the mesh becomes finer, which is a shortcoming of the classic IFE methods in some situations. Trace inequalities are established for both linear and bilinear IFE functions that are not only critical for the error analysis of these new IFE methods but are also of a great potential to be useful in error analysis for other...
200 citations
TL;DR: In this article, an optimal parabolic contour is selected on the basis of the distance and the strength of the singularities of the Laplace transform, with the aim of minimizing the computational effort and reducing the propagation of errors.
Abstract: The Mittag-Leffler (ML) function plays a fundamental role in fractional calculus but very few methods are available for its numerical evaluation. In this work we present a method for the efficient computation of the ML function based on the numerical inversion of its Laplace transform (LT): an optimal parabolic contour is selected on the basis of the distance and the strength of the singularities of the LT, with the aim of minimizing the computational effort and reducing the propagation of errors. Numerical experiments are presented to show accuracy and efficiency of the proposed approach. The application to the three parameter ML (also known as Prabhakar) function is also presented.
197 citations
TL;DR: This paper shows that Anderson is locally r-linearly convergent if the fixed point map is a contraction and the coefficients in the linear combination remain bounded and proves q-linear convergence of Anderson(1) and, in the case of linear problems, Anderson($m$).
Abstract: Anderson($m$) is a method for acceleration of fixed point iteration which stores m+1 prior evaluations of the fixed point map and computes the new iteration as a linear combination of those evaluations. Anderson(0) is fixed point iteration. In this paper we show that Anderson($m$) is locally r-linearly convergent if the fixed point map is a contraction and the coefficients in the linear combination remain bounded. Without assumptions on the coefficients, we prove q-linear convergence of Anderson(1) and, in the case of linear problems, Anderson($m$). We observe that the optimization problem for the coefficients can be formulated and solved in nonstandard ways and report on numerical experiments which illustrate the ideas.
182 citations
TL;DR: Estimates of flux a posteriori error estimates for conforming, nonconforming, discontinuous Galerkin, and mixed finite element discretizations of the two-dimensional Poisson problem are guaranteed, locally computable, locally efficient, and robust with respect to polynomial degree.
Abstract: We present equilibrated flux a posteriori error estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed finite element discretizations of the two-dimensional Poisson problem. Relying on the equilibration by the mixed finite element solution of patchwise Neumann problems, the estimates are guaranteed, locally computable, locally efficient, and robust with respect to polynomial degree. Maximal local overestimation is guaranteed as well. Numerical experiments suggest asymptotic exactness for the incomplete interior penalty discontinuous Galerkin scheme.
163 citations
TL;DR: A robust and efficient time integrator for dynamical tensor approximation in the tensor train or matrix product state format is presented and numerical experiments with problems from quantum molecular dynamics and with iterative processes in the Tensor train format are included.
Abstract: A robust and efficient time integrator for dynamical tensor approximation in the tensor train or matrix product state format is presented. The method is based on splitting the projector onto the tangent space of the tensor manifold. The algorithm can be used for updating time-dependent tensors in the given data-sparse tensor train/matrix product state format and for computing an approximate solution to high-dimensional tensor differential equations within this data-sparse format. The formulation, implementation, and theoretical properties of the proposed integrator are studied, and numerical experiments with problems from quantum molecular dynamics and with iterative processes in the tensor train format are included.
138 citations
TL;DR: The stability and error estimates of the local discontinuous Galerkin (LDG) methods coupled with carefully chosen implicit-explicit (IMEX) Runge--Kutta time discretization up to third order accuracy for solving one-dimensional linear advection-diffusion equations are analyzed.
Abstract: The main purpose of this paper is to analyze the stability and error estimates of the local discontinuous Galerkin (LDG) methods coupled with carefully chosen implicit-explicit (IMEX) Runge--Kutta time discretization up to third order accuracy for solving one-dimensional linear advection-diffusion equations. In the time discretization the advection term is treated explicitly and the diffusion term implicitly. There are three highlights of this work. The first is that we establish an important relationship between the gradient and interface jump of the numerical solution with the independent numerical solution of the gradient in the LDG methods. The second is that, by aid of the aforementioned relationship and the energy method, we show that the IMEX LDG schemes are unconditionally stable for the linear problems in the sense that the time-step $\tau$ is only required to be upper-bounded by a constant which depends on the ratio of the diffusion and the square of the advection coefficients and is independent...
110 citations
TL;DR: In this paper, a mixed finite element method for a modified Cahn-Hilliard equation coupled with a nonsteady Darcy-Stokes flow was devised and analyzed for phase separation and coupled fluid flow in immiscible binary fluids and diblock copolymer melts.
Abstract: In this paper we devise and analyze a mixed finite element method for a modified Cahn--Hilliard equation coupled with a nonsteady Darcy--Stokes flow that models phase separation and coupled fluid flow in immiscible binary fluids and diblock copolymer melts. The time discretization is based on a convex splitting of the energy of the equation. We prove that our scheme is unconditionally energy stable with respect to a spatially discrete analogue of the continuous free energy of the system and unconditionally uniquely solvable. We prove that the discrete phase variable is bounded in $L^\infty \left(0,T;L^\infty\right)$ and the discrete chemical potential is bounded in $L^\infty \left(0,T;L^2\right)$, 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.
86 citations
TL;DR: An iterative/recursive algorithm is studied for recovering unknown sources of acoustic field with multifrequency measurement data and the first convergence result toward multifrequency inverse source problems is obtained by assuming the background medium is homogeneous and the measurement data is noise-free.
Abstract: An iterative/recursive algorithm is studied for recovering unknown sources of acoustic field with multifrequency measurement data. Under additional regularity assumptions on source functions, the first convergence result toward multifrequency inverse source problems is obtained by assuming the background medium is homogeneous and the measurement data is noise-free. Error estimates are also provided when the observation data is contaminated by noise. Numerical examples verify the reliability and efficiency of our proposed algorithm.
85 citations
TL;DR: It is shown that if the pollution error of the CIP-FEM is sufficiently small, then the pollution errors of both methods in $H^1$-norm are bounded by $O(k^{2p+1}h 2p})$, which coincides with the phaseerror of the FEM obtained by existent dispersion analyses on Cartesian grids.
Abstract: A preasymptotic error analysis of the finite element method (FEM) and some continuous interior penalty finite element method (CIP-FEM) for the Helmholtz equation in two and three dimensions is proposed. $H^1$- and $L^2$-error estimates with explicit dependence on the wave number $k$ are derived. In particular, it is shown that if $k^{2p+1}h^{2p}$ is sufficiently small, then the pollution errors of both methods in $H^1$-norm are bounded by $O(k^{2p+1}h^{2p})$, which coincides with the phase error of the FEM obtained by existent dispersion analyses on Cartesian grids, where $h$ is the mesh size, and $p$ is the order of the approximation space and is fixed. The CIP-FEM extends the classical one by adding more penalty terms on jumps of higher (up to $p$th order) normal derivatives in order to reduce efficiently the pollution errors of higher order methods. Numerical tests are provided to verify the theoretical findings and to illustrate the great capability of the CIP-FEM in reducing the pollution effect.
84 citations
TL;DR: Enriching the FETI-DP coarse space by a few numerically computed eigenvectors yields independence of the contrast of the coefficients even in challenging situations.
Abstract: A coarse space is constructed for the dual-primal finite element tearing and interconnecting (FETI-DP) domain decomposition method applied to highly heterogeneous problems by solving local generalized eigenvalue problems For certain problems with highly varying coefficients, eg, from multiscale simulations, the coefficient jump will appear in the condition number bound even if standard techniques such as scaling and the weighting of constraints are used The FETI-DP theory is revisited and two central estimates are identified where the dependency on the coefficient contrast can enter the condition number bound The first is a Poincare inequality and the second an extension theorem These estimates are replaced by local eigenvalue problems Enriching the FETI-DP coarse space by a few numerically computed eigenvectors yields independence of the contrast of the coefficients even in challenging situations
TL;DR: The paper discusses the construction and properties of implicit-explicit and implicit-implicit methods in the new framework, which introduces additional flexibility when compared to traditional partitioned Runge--Kutta formalism and therefore offers additional opportunities for the development of flexible solvers for systems with multiple scales, or driven by multiple physical processes.
Abstract: This work considers a general structure of the additively partitioned Runge--Kutta methods by allowing for different stage values as arguments of different components of the right-hand side. An order conditions theory is developed for the new formulation of generalized additive methods, and stability and monotonicity investigations are carried out. The paper discusses the construction and properties of implicit-explicit and implicit-implicit methods in the new framework. The new approach, named GARK, introduces additional flexibility when compared to traditional partitioned Runge--Kutta formalism and therefore offers additional opportunities for the development of flexible solvers for systems with multiple scales, or driven by multiple physical processes.
TL;DR: A new characterization of sufficient conditions for the Lie-Trotter splitting to capture the numerical invariant measure of nonlinear ergodic Langevin dynamics up to an arbitrary order is discussed.
Abstract: A new characterization of sufficient conditions for the Lie-Trotter splitting to capture the numerical invariant measure of nonlinear ergodic Langevin dynamics up to an arbitrary order is discussed. Our characterization relies on backward error analysis and needs weaker assumptions than assumed so far in the literature. In particular, neither high weak order of the splitting scheme nor symplecticity are necessary to achieve high order approximation of the invariant measure of the Langevin dynamics. Numerical experiments confirm our theoretical findings.
TL;DR: The convergence of the approximate solutions is proved, providing the existence of a solution in a slightly more general setting than in other results in the current literature.
Abstract: We present a Lax--Friedrichs-type algorithm to numerically integrate a class of nonlocal and nonlinear systems of conservation laws in several space dimensions. The convergence of the approximate solutions is proved, also providing the existence of a solution in a slightly more general setting than in other results in the current literature. An application to a crowd dynamics model is considered.
TL;DR: A two-grid block-centered finite difference method is proposed for solving the two-dimensional Darcy--Forchheimer model describing non-Darcy flow in porous media and optimal order error estimates for pressure and velocity in discrete $L^2$ norms are obtained.
Abstract: A two-grid block-centered finite difference method is proposed for solving the two-dimensional Darcy--Forchheimer model describing non-Darcy flow in porous media. To construct the two-grid method we modify the original nonlinear elliptic operator of Darcy--Forchheimer flow to a twice continuously differentiable one by introducing a small and positive parameter $\varepsilon$. By using the two-grid method, solving a nonlinear equation on a fine grid is reduced to solving a nonlinear equation on a coarse grid together with solving a linear equation on a fine grid. Optimal order error estimates for pressure and velocity in discrete $L^2$ norms are obtained. Some numerical examples are given to show the accuracy and efficiency of the presented method.
TL;DR: Under some regularity assumptions imposed on the true solution, optimal order error estimates are derived for the linear element solution and this theoretical result is illustrated numerically.
Abstract: In this paper a fully dynamic viscoelastic contact problem is studied. The contact is assumed to be bilateral and frictional, where the friction law is described by a nonmonotone relation between the tangential stress and the tangential velocity. A weak formulation of the problem leads to a second order nonmonotone subdifferential inclusion, also known as a second order hyperbolic hemivariational inequality. We study both semidiscrete and fully discrete approximation schemes and bound the errors of the approximate solutions. Under some regularity assumptions imposed on the true solution, optimal order error estimates are derived for the linear element solution. This theoretical result is illustrated numerically.
TL;DR: In this article, an approximation method for advection-diffusion-reaction equa-tions where the (generalized) degrees of freedom are polynomials of order k>=0 at mesh faces is presented.
Abstract: We design and analyze an approximation method for advection-diffusion-reaction equa-tions where the (generalized) degrees of freedom are polynomials of order k>=0 at mesh faces. The method hinges on local discrete reconstruction operators for the diffusive and advective derivatives and a weak enforcement of boundary conditions. Fairly general meshes with poly-topal and nonmatching cells are supported. Arbitrary polynomial orders can be considered, including the case k=0 which is closely related to Mimetic Finite Difference/Mixed-Hybrid Finite Volume methods. The error analysis covers the full range of Peclet numbers, including the delicate case of local degeneracy where diffusion vanishes on a strict subset of the domain. Computational costs remain moderate since the use of face unknowns leads to a compact stencil with reduced communications. Numerical results are presented.
TL;DR: In this article, a new formulation for the finite element immersed boundary method which makes use of a distributed Lagrange multiplier is introduced. But it is not shown that the model is stable with respect to the time step size.
Abstract: We introduce a new formulation for the finite element immersed boundary method which makes use of a distributed Lagrange multiplier. We prove that a full discretization of our model, based on a semi-implicit time advancing scheme, is unconditionally stable with respect to the time step size.
TL;DR: A preconditioned version of the Douglas--Rachford splitting method for solving convex-concave saddle-point problems associated with Fenchel--Rockafellar duality is proposed and weak convergence in Hilbert space under minimal assumptions is proved.
Abstract: We propose a preconditioned version of the Douglas--Rachford splitting method for solving convex-concave saddle-point problems associated with Fenchel--Rockafellar duality. Our approach makes it possible to use approximate solvers for the linear subproblem arising in this context. We prove weak convergence in Hilbert space under minimal assumptions. In particular, various efficient preconditioners are introduced in this framework for which only a few inner iterations are needed instead of computing an exact solution or controlling the error. The method is applied to a discrete total-variation denoising problem. Numerical experiments show that the proposed algorithms with appropriate preconditioners are very competitive with existing fast algorithms including the first-order primal-dual algorithm for saddle-point problems of Chambolle and Pock.
TL;DR: The stability and convergence results presented herein provide the first rigorous justification of the applicability of cell-centered finite volume methods to problems in linear elasticity.
Abstract: We show convergence of a cell-centered finite volume discretization for linear elasticity. The discretization, termed the MPSA method, was recently proposed in the context of geological applications, where cell-centered variables are often preferred. Our analysis utilizes a hybrid variational formulation, which has previously been used to analyze finite volume discretizations for the scalar diffusion equation. The current analysis deviates significantly from the previous in three respects. First, additional stabilization leads to a more complex saddle-point problem. Second, a discrete Korn's inequality has to be established for the global discretization. Finally, robustness with respect to the Poisson ratio is analyzed. The stability and convergence results presented herein provide the first rigorous justification of the applicability of cell-centered finite volume methods to problems in linear elasticity.
TL;DR: It is proved that under some suitable initial and boundary discretizations, the (2k+1)th order superconvergence rate of the DG approximation at the downwind points and for the cell averages, when piecewise tensor-product polynomials of degree k are used.
Abstract: This paper is concerned with superconvergence properties of discontinuous Galerkin (DG) methods for two-dimensional linear hyperbolic conservation laws over rectangular meshes when upwind fluxes are used. We prove, under some suitable initial and boundary discretizations, the (2k+1)th order superconvergence rate of the DG approximation at the downwind points and for the cell averages, when piecewise tensor-product polynomials of degree k are used. Moreover, we prove that the gradient of the DG solution is superconvergent with a rate of (k+1)th order at all interior left Radau points; and the function value approximation is superconvergent at all right Radau points with a rate of (k+2)th order. Numerical experiments indicate that the aforementioned superconvergence rates are sharp.
TL;DR: Finite element approximations of the two-dimensional and three-dimensional Signorini problems with linear and quadratic finite elements are considered and an optimal error bound is obtained.
Abstract: The basic $H^1$-finite element error estimate of order $h$ with only $H^2$-regularity on the solution has not been yet established for the simplest two-dimensional Signorini problem approximated by a discrete variational inequality (or the equivalent mixed method) and linear finite elements. To obtain an optimal error bound in this basic case and also when considering more general cases (e.g., the three-dimensional problem, quadratic finite elements), additional assumptions on the exact solution (in particular on the unknown contact set, see [Z. Belhachmi and F. Ben Belgacem, Math. Comp., 72 (2003), pp. 83--104; S. Hueber and B. Wohlmuth, SIAM J. Numer. Anal., 43 (2005), pp. 156--173; B. Wohlmuth, A. Popp, M. Gee, and W. Wall, Comput. Mech. 49 (2012), pp. 735--747] had to be used. In this paper, we consider finite element approximations of the two-dimensional and three-dimensional Signorini problems with linear and quadratic finite elements. In the analysis, we remove all the additional assumptions and pr...
TL;DR: A priori error estimates in the energy norm for certain fluxes are derived and numerical experiments showing that optimal convergence in $L^2$ is obtained are presented.
Abstract: We develop and analyze a new strategy for the spatial discontinuous Galerkin discretization of wave equations in second-order form. The method features a direct, mesh-independent approach to defini...
TL;DR: An error analysis of trigonometric integrators (or exponential integrators) applied to spatial semi-discretizations of semilinear wave equations with periodic boundary conditions in one space dimension shows optimal second-order convergence requiring only that the exact solution is of finite energy.
Abstract: An error analysis of trigonometric integrators (or exponential integrators) applied to spatial semidiscretizations of semilinear wave equations with periodic boundary conditions in one space dimension is given. In particular, optimal second-order convergence is shown requiring only that the exact solution is of finite energy. The analysis is uniform in the spatial discretization parameter. It covers the impulse method which coincides with the method of Deuflhard and the mollified impulse method of Garcia-Archilla, Sanz-Serna, and Skeel as well as the trigonometric methods proposed by Hairer and Lubich and by Grimm and Hochbruck. The analysis can also be used to explain the convergence behavior of the Stormer--Verlet/leapfrog time discretization.
TL;DR: This work develops new implicit-explicit time integrators based on two-step Runge--Kutta methods that offers extreme flexibility in the construction of partitioned integrators since no coupling conditions are necessary.
Abstract: This work develops new implicit-explicit time integrators based on two-step Runge--Kutta methods. The class of schemes of interest is characterized by linear invariant preservation and high stage orders. Theoretical consistency, stability, and stiff convergence analyses are performed to reveal the excellent properties of these methods. The new framework offers extreme flexibility in the construction of partitioned integrators since no coupling conditions are necessary. Practical schemes of orders three, four, and six are constructed and are used to solve several test problems. Numerical results confirm the theoretical findings.
TL;DR: In this article, the optimal error estimates for spectral Petrov-Galerkin methods and spectral collocation methods for linear fractional ordinary differential equations with initial value on a finite interval are presented.
Abstract: We present optimal error estimates for spectral Petrov--Galerkin methods and spectral collocation methods for linear fractional ordinary differential equations with initial value on a finite interval. We also develop Laguerre spectral Petrov--Galerkin methods and collocation methods for fractional equations on the half line. Numerical results confirm the error estimates.
TL;DR: A family of second-order semi-implicit time integration methods for systems of ordinary differential equations (ODEs) with stiff damping term that are capable of exactly preserving the steady states as well as maintaining the sign of the computed solution under the time step restriction determined by the nonstiff part of the system only.
Abstract: In this paper, we develop a family of second-order semi-implicit time integration methods for systems of ordinary differential equations (ODEs) with stiff damping term. The important feature of the new methods resides in the fact that they are capable of exactly preserving the steady states as well as maintaining the sign of the computed solution under the time step restriction determined by the nonstiff part of the system only. The new semi-implicit methods are based on the modification of explicit strong stability preserving Runge--Kutta (SSP-RK) methods and are proven to have a formal second order of accuracy, $A(\alpha)$-stability, and stiff decay. We illustrate the performance of the proposed SSP-RK based semi-implicit methods on both a scalar ODE example and a system of ODEs arising from the semi-discretization of the shallow water equations with stiff friction term. The obtained numerical results clearly demonstrate that the ability of the introduced ODE solver to exactly preserve equilibria plays ...
TL;DR: A smoothing and an approximation property are used to prove uniform convergence of the W-cycle scheme with respect to the discretization parameters and the number of levels, provided theNumber of smoothing steps is chosen of order $p^{2+\mu}$.
Abstract: We present W-cycle $h$-, $p$-, and $hp$-multigrid algorithms for the solution of the linear system of equations arising from a wide class of $hp$-version discontinuous Galerkin discretizations of elliptic problems Starting from a classical framework in geometric multigrid analysis, we define a smoothing and an approximation property, which are used to prove uniform convergence of the W-cycle scheme with respect to the discretization parameters and the number of levels, provided the number of smoothing steps is chosen of order $p^{2+\mu}$, where $p$ is the polynomial approximation degree and $\mu=0,1$ A discussion on the effects of employing inherited or noninherited sublevel solvers is also presented Numerical experiments confirm the theoretical results
TL;DR: In this article, a numerical discretization of the one-dimensional aggregation equation was proposed for which the convergence towards duality solutions of the aggregation equation is proved, based on a careful choice of the discretized macroscopic velocity $v$ in order to give a sense to the product $v \rho$.
Abstract: We focus in this work on the numerical discretization of the one-dimensional aggregation equation $\partial_t\rho + \partial_x (v\rho)=0$, $v=a(W'*\rho)$, in the attractive case. Finite time blow up of smooth initial data occurs for potential $W$ having a Lipschitz singularity at the origin. A numerical discretization is proposed for which the convergence towards duality solutions of the aggregation equation is proved. It relies on a careful choice of the discretized macroscopic velocity $v$ in order to give a sense to the product $v \rho$. Moreover, using the same idea, we propose an asymptotic preserving scheme for a kinetic system in hyperbolic scaling converging towards the aggregation equation in the hydrodynamic limit. Finally numerical simulations are provided to illustrate the results.
TL;DR: It is proved that algebraic convergence rates can be achieved for the $h$-$p$ version of the CPG method with quasi-uniform meshes and exponential rates of convergence can be achieve for solutions with start-up singularities by using geometric time partitions and linearly increasing polynomial degrees.
Abstract: We present an $h$-$p$ version of the continuous Petrov--Galerkin (CPG) finite element method for linear Volterra integro-differential equations with smooth and nonsmooth kernels. We establish a priori error estimates in the $L^2$-, $H^1$-, and $L^\infty$-norms that are completely explicit with respect to the local discretization and regularity parameters. For singular solutions caused by the weakly singular kernels, we prove that algebraic convergence rates can be achieved for the $h$-$p$ version of the CPG method with quasi-uniform meshes. Moreover, we show that exponential rates of convergence can be achieved for solutions with start-up singularities by using geometric time partitions and linearly increasing polynomial degrees. Numerical experiments are provided to illustrate the theoretical results.