Related papers: Combining the DPG method with finite elements
Generalizing the framework of an ultra-weak formulation for a hypersingular integral equation on closed polygons in [N. Heuer, F. Pinochet, arXiv 1309.1697 (to appear in SIAM J. Numer. Anal.)], we study the case of a hypersingular integral…
In this paper, we propose a weak Galerkin finite element method (WG) for solving singularly perturbed convection-diffusion problems on a Bakhvalov-type mesh in 2D. Our method is flexible and allows the use of discontinuous approximation…
We introduce the concept of data-driven finite element methods. These are finite-element discretizations of partial differential equations (PDEs) that resolve quantities of interest with striking accuracy, regardless of the underlying mesh…
In this paper, we describe a stable finite element formulation for advection-diffusion-reaction problems that allows for robust automatic adaptive strategies to be easily implemented. We consider locally vanishing, heterogeneous, and…
A generalized finite element method is proposed for solving a heterogeneous reaction-diffusion equation with a singular perturbation parameter $\varepsilon$, based on locally approximating the solution on each subdomain by solution of a…
At the fully discrete setting, stability of the discontinuous Petrov--Galerkin (DPG) method with optimal test functions requires local test spaces that ensure the existence of Fortin operators. We construct such operators for $H^1$ and…
This article establishes a discrete maximum principle (DMP) for the approximate solution of convection-diffusion-reaction problems obtained from the weak Galerkin finite element method on nonuniform rectangular partitions. The DMP analysis…
In the numerical solution of partial differential equations (PDEs), a central question is the one of building variational formulations that are inf-sup stable not only at the infinite-dimensional level, but also at the finite-dimensional…
We present a general family of subcell limiting strategies to construct robust high-order accurate nodal discontinuous Galerkin (DG) schemes. The main strategy is to construct compatible low order finite volume (FV) type discretizations…
In this paper, we propose and analyze a high-order finite volume method for the Poisson problem based on the reduced discontinuous Galerkin (RDG) space. The main idea is to employ the RDG space as the trial space and the piecewise constant…
This paper develops and analyzes a class of semi-discrete and fully discrete weak Galerkin finite element methods for unsteady incompressible convective Brinkman-Forchheimer equations. For the spatial discretization, the methods adopt the…
In this paper, we present convergence analysis of high-order finite element based methods, in particular, we focus on a discontinuous Galerkin scheme using summation-by-parts operators. To this end, it is crucial that structure preserving…
The mixed finite element method for the Poisson problem with the Raviart-Thomas elements of low-level can be interpreted as a finite volume method with a non-local gradient. In this contribution, we propose a variant of Petrov-Galerkin type…
This article devises a new primal-dual weak Galerkin finite element method for the convection-diffusion equation. Optimal order error estimates are established for the primal-dual weak Galerkin approximations in various discrete norms and…
This paper analyzes a space-time finite element method for fractional wave problems. The method uses a Petrov-Galerkin type time-stepping scheme to discretize the time fractional derivative of order $ \gamma $ ($1<\gamma<2$). We establish…
In this paper, a new strategy for a sub-element based shock capturing for discontinuous Galerkin (DG) approximations is presented. The idea is to interpret a DG element as a collection of data and construct a hierarchy of low to high order…
This paper introduces new discretization schemes for time-harmonic Maxwell equations in a connected domain by using the weak Galerkin (WG) finite element method. The corresponding WG algorithms are analyzed for their stability and…
This paper presents a numerical method for div-curl systems with normal boundary conditions by using a finite element technique known as primal-dual weak Galerkin (PDWG). The PDWG finite element scheme for the div-curl system has two…
Variational time discretization schemes are getting of increasing importance for the accurate numerical approximation of transient phenomena. The applicability and value of mixed finite element methods (MFEM) in space for simulating…
We consider the initial/boundary value problem for a diffusion equation involving multiple time-fractional derivatives on a bounded convex polyhedral domain. We analyze a space semidiscrete scheme based on the standard Galerkin finite…