Related papers: Hessian Recovery for Finite Element Methods
We present a novel approach for high-order accurate numerical differentiation on unstructured meshes of quadrilateral elements. To differentiate a given function, an auxiliary function with greater smoothness properties is defined which…
We devise and analyze two hybrid high-order (HHO) methods for the numerical approximation of the biharmonic problem. The methods support polyhedral meshes, rely on the primal formulation of the problem, and deliver $O(h^{k+1})$ $H^2$-error…
In this paper, we introduce a Lagrange multiplier approach to construct linearly implicit energy-preserving schemes of arbitrary order for general Hamiltonian PDEs. Unlike the widely used auxiliary variable methods, this novel approach does…
In this paper, we introduce new stable mixed finite elements of any order on polytopal mesh for solving second order elliptic problem. We establish optimal order error estimates for velocity and super convergence for pressure. Numerical…
A new class of high-order maximum principle preserving numerical methods is proposed for solving parabolic equations, with application to the semilinear Allen--Cahn equation. The proposed method consists of a $k$th-order multistep…
We present a one-dimensional high-order moving-mesh finite element method for moving boundary problems where the boundary velocity depends implicitly on the solution in the interior of the domain. The method employs a conservative arbitrary…
We present numerical experiments for geophysics electromagnetic (EM) modeling based upon high-order edge elements and supervised $h+p$ refinement approaches on massively parallel computers. Our high-order $h+p$ refinement strategy is based…
Matrix completion, i.e., the exact and provable recovery of a low-rank matrix from a small subset of its elements, is currently only known to be possible if the matrix satisfies a restrictive structural constraint---known as {\em…
We present a superconvergent hybridizable discontinuous Galerkin (HDG) method for the steady-state incompressible Navier-Stokes equations on general polyhedral meshes. For arbitrary conforming polyhedral mesh, we use polynomials of degree…
In this paper, we develop a straightforward $C^0$ linear finite element method for sixth-order elliptic equations. The basic idea is to use gradient recovery techniques to generate higher-order numerical derivatives from a $C^0$ linear…
The embedded discontinuous Galerkin (EDG) method by Cockburn et al. [SIAM J. Numer. Anal., 2009, 47(4), 2686-2707] is obtained from the hybridizable discontinuous Galerkin method by changing the space of the Lagrangian multiplier from…
We consider within a finite element approach the usage of different adaptively refined meshes for different variables in systems of nonlinear, time-depended PDEs. To resolve different solution behaviours of these variables, the meshes can…
We propose energy-conserving discontinuous Galerkin (DG) methods for symmetric linear hyperbolic systems on general unstructured meshes. Optimal a priori error estimates of order $k+1$ are obtained for the semi-discrete scheme in one…
This work studies the problem of maximizing a higher degree real homogeneous multivariate polynomial over the unit sphere. This problem is equivalent to finding the leading eigenvalue of the associated symmetric tensor of higher order,…
In this work, we investigate the inverse problem of recovering a potential coefficient in an elliptic partial differential equation from the observations at deterministic sampling points in the domain subject to random noise. We employ a…
The classical continuous finite element method with Lagrangian $Q^k$ basis reduces to a finite difference scheme when all the integrals are replaced by the $(k+1)\times (k+1)$ Gauss-Lobatto quadrature. We prove that this finite difference…
In this work, we study advection-robust Hybrid High-Order discretizations of the Oseen equations. For a given integer $k\ge 0$, the discrete velocity unknowns are vector-valued polynomials of total degree $\le k$ on mesh elements and faces,…
We introduce a new class of unfitted finite element methods with high order accurate numerical integration over curved surfaces and volumes which are only implicitly defined by level set functions. An unfitted finite element method which is…
This paper proposes a homogeneous second-order descent framework (HSODF) for nonconvex and convex optimization based on the generalized homogeneous model (GHM). In comparison to the Newton steps, the GHM can be solved by extremal symmetric…
The purpose of this research is to describe an efficient iterative method suitable for obtaining high accuracy solutions to high frequency time-harmonic scattering problems. The method allows for both refinement of local polynomial degree…