Related papers: A Nitsche Hybrid multiscale method with non-matchi…
The elliptic 2-Hessian equation is a fully nonlinear partial differential equation (PDE) that is related to intrinsic curvature for three dimensional manifolds. We introduce two numerical methods for this PDE: the first is provably…
We propose a novel stochastic gradient descent method for solving linear least squares problems with partially observed data. Our method uses submatrices indexed by a randomly selected pair of row and column index sets to update the iterate…
When solving partial differential equations using classical schemes such as finite difference or finite volume methods, sufficiently fine meshes and carefully designed schemes are required to achieve high-order accuracy of numerical…
In this paper, a non-uniform time-stepping convex-splitting numerical algorithm for solving the widely used time-fractional Cahn-Hilliard equation is introduced. The proposed numerical scheme employs the $L1^+$ formula for discretizing the…
This paper studies a low-communication algorithm for solving elliptic partial differential equations (PDE's) on high-performance machines, the nested iteration with range decomposition algorithm (NIRD). Previous work has shown that NIRD…
In this paper we present a one dimensional second order accurate method to solve Elliptic equations with discontinuous coefficients on an arbitrary interface. Second order accuracy for the first derivative is obtained as well. The method is…
For the optimal control problem governed by elliptic equations with interfaces, we present a numerical method based on the Hansbo's Nitsche-XFEM. We followed the Hinze's variational discretization concept to discretize the continuous…
We develop a higher regularity theory for general quasilinear elliptic equations and systems in divergence form with random coefficients. The main result is a large-scale $L^\infty$-type estimate for the gradient of a solution. The estimate…
We establish a microscopic convexity principle for nonlinear elliptic and parabolic partial differential equations in general form.
A new method for solving numerically stochastic partial differential equations (SPDEs) with multiple scales is presented. The method combines a spectral method with the heterogeneous multiscale method (HMM) presented in [W. E, D. Liu, and…
The Aubin--Nitsche trick is a common tool to show $L^2$-error estimates for discretizations of $H^1$-elliptic linear partial differential equations arising for example as Euler--Lagrange equations of a quadratic energy functional. The…
We present a hybridization technique for summation-by-parts finite difference methods with weak enforcement of interface and boundary conditions for second order, linear elliptic partial differential equations. The method is based on…
In this work, we propose a high-order multiscale method for an elliptic model problem with rough and possibly highly oscillatory coefficients. Convergence rates of higher order are obtained using the regularity of the right-hand side only.…
We present a new composite mesh finite element method for fluid--structure interaction problems. The method is based on surrounding the structure by a boundary-fitted fluid mesh which is embedded into a fixed background fluid mesh. The…
In this work, we develop an efficient solver based on neural networks for second-order elliptic equations with variable coefficients and singular sources. This class of problems covers general point sources, line sources and the combination…
We develop an efficient reduced basis method for the frictional contact problem formulated using Nitsche's method. We focus on the regime of small deformations and on Tresca friction. The key idea ensuring the computational efficiency of…
We propose a multiscale approach for a nonlinear Helmholtz problem with possible oscillations in the Kerr coefficient, the refractive index, and the diffusion coefficient. The method does not rely on structural assumptions on the…
In this paper, we consider a new approach for semi-discretization in time and spatial discretization of a class of semi-linear stochastic partial differential equations (SPDEs) with multiplicative noise. The drift term of the SPDEs is only…
This paper presents the development and analysis of an asymptotically compatible (AC) unfitted finite element method for one-dimensional nonlocal elliptic interface problems. The proposed method achieves optimal error estimates through…
We obtain a local estimate for the gradient of solutions to a second-order elliptic equation in divergence form with bounded measurable coefficients that are square-Dini continuous at the single point x=0. In particular, we treat the case…