SIAM Journal on Numerical Analysis

In this paper, we develop a provably energy stable and conservative discontinuous spectral element method for the shifted wave equation in second order form. The proposed method combines the advantages and central ideas of the following very successful ...

In this paper, we investigate the problem of reconstructing the historical distribution for a nonlinear diffusion equation, in which the diffusion is driven by not only a nonlocal operator but also a locally one. The problem naturally arises in many real-...

We are interested in the Euler--Maruyama discretization of a stochastic differential equation in dimension $d$ with constant diffusion coefficient and bounded measurable drift coefficient. In the scheme, a randomization of the time variable is used to get ...

In this paper we shall establish an a priori L$^2$-norm error estimate of the fourth order Runge--Kutta discontinuous Galerkin method for solving sufficiently smooth solutions of one-dimensional scalar nonlinear conservation laws. The optimal order of ...

We consider residual-based a posteriori error estimators for Galerkin discretizations of time-harmonic Maxwell's equations. We focus on configurations where the frequency is high, or close to a resonance frequency, and derive reliability and efficiency ...

In this paper, we propose a new family of fully discrete Sinc-$\theta$ schemes for solving backward stochastic differential equations (BSDEs). More precisely, we consider the $\theta$-schemes for the temporal discretizations and then adopt the Sinc ...

In this paper, we analyze two numerical staggered schemes, one fully implicit and the other semi-implicit, for the compressible Navier--Stokes equations with a singular pressure law: a system which degenerates towards the free-congested Navier--Stokes ...

In the present contribution we develop a sharper error analysis for the Virtual Element Method, applied to a model elliptic problem, that separates the element boundary and element interior contributions to the error. As a consequence we are able to ...

Fractional Gaussian noise models the time series with long-range dependence; when the Hurst index $H\in(1/2,1)$, it has positive correlation reflecting a persistent autocorrelation structure. This paper studies the numerical method for solving the ...

The energy dissipation law and the maximum bound principle (MBP) are two important physical features of the well-known Allen--Cahn equation. While some commonly used first-order time stepping schemes have turned out to preserve unconditionally both the ...

Several div-conforming and divdiv-conforming finite elements for symmetric tensors on simplexes in arbitrary dimension are constructed in this work. The shape function space is first split as the trace space and the bubble space. The later is further ...

We establish improved uniform error bounds on a second-order Strang time-splitting method which is equivalent to an exponential wave integrator for the long-time dynamics of the nonlinear Klein--Gordon equation (NKGE) with weak cubic nonlinearity, whose ...

Numerically solving the 2D Helmholtz equation is widely known to be very difficult largely due to its highly oscillatory solution, which brings about the pollution effect. A very fine mesh size is necessary to deal with a large wavenumber leading to a ...

Identifying hidden dynamics from observed data is a significant and challenging task in a wide range of applications. Recently, the combination of linear multistep methods (LMMs) and deep learning has been successfully employed to discover dynamics, ...

Many nonlocal models have adopted Euclidean balls as the nonlocal interaction neighborhoods. When solving them numerically, it is sometimes convenient to adopt polygonal approximations of such balls. A crucial question is to what extent such ...

We propose a new discretization method for PDEs on moving domains in the setting of unfitted finite element methods, which is provably higher-order accurate in space and time. In the considered setting, the physical domain that evolves essentially ...

We consider the Helmholtz equation with a nonconstant coefficient, defined in unbounded domains external to 2D bounded ones, endowed with a Dirichlet condition on the boundary and the Sommerfeld radiation condition at infinity. To solve it, we reduce the ...

We consider the numerical approximation of $\mathbb{P}[G\in \Omega]$, where the $d$-dimensional random variable $G$ cannot be sampled directly, but there is a hierarchy of increasingly accurate approximations $\{G_\ell\}_{\ell\in\mathbb{N}}$ which can be ...

We present a fully discrete convergent finite difference scheme for the Q-tensor flow of liquid crystals based on the energy-stable semidiscrete scheme by Zhao et al. [Comput. Methods Appl. Mech. Engrg., 2017, pp. 803--825]. We prove stability properties ...

We propose a fourth-order unfitted characteristic finite element method to solve the advection-diffusion equation on time-varying domains. Based on a characteristic-Galerkin formulation, our method combines the cubic MARS (Mapping and Adjusting Regular ...

In this paper, we propose an infinite-dimensional version of the Stein variational gradient descent (iSVGD) method for solving Bayesian inverse problems. The method can generate approximate samples from posteriors efficiently. Based on the concepts of ...

We prove that the two-step backward differentiation formula (BDF) method is stable on arbitrary time grids; while the variable-step three-step backward differentiation formula scheme is stable if almost all adjacent step ratios are less than 2.553. These ...

A new type of low-regularity integrator is proposed for the Navier--Stokes equations. Unlike the other low-regularity integrators for nonlinear dispersive equations, which are all fully explicit in time, the proposed method is a semi-implicit exponential ...

Uniform convergence of the geometric multigrid V-cycle is proven for hybridized discontinuous Galerkin methods with a new set of assumptions on the injection operators from coarser to finer meshes. The scheme involves standard smoothers and local solvers ...

In this paper, we design and analyze a hybrid high-order approximation for a class of quasilinear elliptic problems of nonmonotone type. The proposed method has several advantages; for instance, it supports an arbitrary order of approximation and general ...