Related papers: A saturation property for the spectral-Galerkin ap…
Building on existing $hp$-adaptive algorithms driven by equilibrated-flux estimators from [ESAIM Math. Model. Numer. Anal. 57 (2023), 329--366] and the references therein, we propose a novel $h$-adaptive algorithm for a fixed polynomial…
We study the exponent of the exponential rate of convergence in terms of the number of degrees of freedom for various non-standard {$p$-version} finite element spaces employing reduced cardinality basis. More specifically, we show that…
We introduce a new stabilization for discontinuous Galerkin methods for the Poisson problem on polygonal meshes, which induces optimal convergence rates in the polynomial approximation degree $p$. In the setting of [S. Bertoluzza and D.…
We apply the discontinuous Galerkin finite element method with a degree $p$ polynomial basis to the linear advection equation and derive a PDE which the numerical solution solves exactly. We use a Fourier approach to derive polynomial…
We introduce an $hp$-version symmetric interior penalty discontinuous Galerkin finite element method (DGFEM) for the numerical approximation of the biharmonic equation on general computational meshes consisting of polygonal/polyhedral…
This paper presents an $hp$ a posteriori error analysis for the 2D Helmholtz equation that is robust in the polynomial degree $p$ and the wave number $k$. For the discretization, we consider a discontinuous Galerkin formulation that is…
We present the analysis for an $hp$ weak Galerkin-FEM for singularly perturbed reaction-convection-diffusion problems in one-dimension. Under the analyticity of the data assumption, we establish robust exponential convergence, when the…
A $p$-adaptive discontinuous Galerkin time-domain method is developed to obtain high-order solutions to electromagnetic scattering problems. A novel feature of the proposed method is the use of divergence error to drive the $p$-adaptive…
In $d$ dimensions, approximating an arbitrary function oscillating with frequency $\lesssim k$ requires $\sim k^d$ degrees of freedom. A numerical method for solving the Helmholtz equation (with wavenumber $k$) suffers from the pollution…
We consider approximation of the variable-coefficient Helmholtz equation in the exterior of a Dirichlet obstacle using perfectly-matched-layer (PML) truncation; it is well known that this approximation is exponentially accurate in the PML…
Time-stepping $hp$-versions discontinuous Galerkin (DG) methods for the numerical solution of fractional subdiffusion problems of order $-\alpha$ with $-1<\alpha<0$ will be proposed and analyzed. Generic $hp$-version error estimates are…
We study the $h$- and $p$-versions of non-conforming harmonic virtual element methods (VEM) for the approximation of the Dirichlet-Laplace problem on a 2D polygonal domain, providing quasi-optimal error bounds. Harmonic VEM do not make use…
In $d$ dimensions, accurately approximating an arbitrary function oscillating with frequency $\lesssim k$ requires $\sim k^d$ degrees of freedom. A numerical method for solving the Helmholtz equation (with wavenumber $k$ and in $d$…
Numerically solving high-dimensional random parametric PDEs poses a challenging computational problem. It is well-known that numerical methods can greatly benefit from adaptive refinement algorithms, in particular when functional…
For the spectral fractional diffusion operator of order $2s\in (0,2)$ in bounded, curvilinear polygonal domains $\Omega$ we prove exponential convergence of two classes of $hp$ discretizations under the assumption of analytic data, without…
This paper generalizes the non-conforming FEM of Crouzeix and Raviart and its fundamental projection property by a novel mixed formulation for the Poisson problem based on the Helmholtz decomposition. The new formulation allows for ansatz…
We study superconvergence property of the linear discontinuous Galerkin finite element method with the polynomial preserving recovery (PPR) and Richardson extrapolation for the two dimensional Helmholtz equation. The error estimate with…
We consider a linear elliptic partial differential equation (PDE) with a generic uniformly bounded parametric coefficient. The solution to this PDE problem is approximated in the framework of stochastic Galerkin finite element methods. We…
It is shown that if the sequence $(p_j(x))$ increases uniformly to $p(x)$ in a bounded, smooth domain $\Omega$, then the sequence $(u_i)$ of solutions to the Dirichlet problem for the $p_i(x)$-Laplacian with fixed boundary datum $\varphi$…
The subject of this work is an adaptive stochastic Galerkin finite element method for parametric or random elliptic partial differential equations, which generates sparse product polynomial expansions with respect to the parametric…