SIAM Journal on Numerical Analysis

This article is about both approximation theory and the numerical solution of partial differential equations (PDEs). First we introduce the notion of reciprocal-log or log-lightning approximation of analytic functions with branch point singularities at ...

Convergence of Dziuk's fully discrete linearly implicit scheme which was proposed in [G. Dziuk, Math. Models Methods Appl. Sci., 4 (1994), pp. 589--606] for curve shortening flow still remains open. In this paper, we prove that this scheme has an optimal ...

Nonintrusive shadowing algorithms efficiently compute $v$, the difference between shadowing trajectories, and then use $v$ to compute derivatives of averaged objectives of chaos with respect to parameters of the dynamical system. However, previous proofs ...

In this article, we consider the stochastic Cahn--Hilliard equation driven by additive noise. We discretize the equation by exploiting the spectral Galerkin method in space and a temporal accelerated implicit Euler method. Based on optimal regularity ...

Consider the elastic scattering of an incident wave by a rigid obstacle in three dimensions, which is formulated as an exterior problem for the Navier equation. By constructing a Dirichlet-to-Neumann (DtN) operator and introducing a transparent boundary ...

We construct high-order semidiscrete-in-time, and fully discrete (with Fourier--Galerkin in space) schemes for the incompressible Navier--Stokes equations with periodic boundary conditions and carry out corresponding error analysis. The schemes are ...

In this work, new finite difference schemes are presented for dealing with the upper-convected time derivative in the context of the generalized Lie derivative. The upper-convected time derivative, which is usually encountered in the constitutive equation ...

Calculating averages with respect to probability measures on submanifolds is often necessary in various application areas such as molecular dynamics, computational statistical mechanics and Bayesian statistics. In recent years, various numerical schemes ...

We consider approximation rates of sparsely connected deep rectified linear unit (ReLU) and rectified power unit (RePU) neural networks for functions in Besov spaces $B^\alpha_{q}(L^p)$ in arbitrary dimension $d$, on general domains. We show that deep rectifier ...

In this paper, we analyze the error estimate of the unconditionally stable and decoupled linear positivity-preserving finite element method (FEM) for solving the chemotaxis-Stokes equations. First, via smoothing the step function in the equation, we study ...

For a diffusion problem modeled by the $p$-Laplacian operator, we are interested in obtaining numerically the two-phase material which maximizes the internal energy. We assume that the amount of the best material is limited. In the framework of a relaxed ...

This is a corrigendum to the article “Error estimates for Galerkin approximations of the Serre equations” by D. C. Antonopoulos, V. A. Dougalis, and D. E. Mitsotakis, SIAM J. Numer. Anal., 55 (2017), pp. 841--868.