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We present scalable iterative solvers and preconditioning strategies for Hybridizable Discontinuous Galerkin (HDG) discretizations of partial differential equations (PDEs) on graphics processing units (GPUs). The HDG method is implemented…
In this paper, we propose a linearized finite element method (FEM) for solving the cubic nonlinear Schr\"{o}dinger equation with wave operator. In this method, a modified leap-frog scheme is applied for time discretization and a Galerkin…
This paper presents a framework for the analysis of discretization methods based on the decomposition into local and global problems. We apply the framework to provide a comprehensive error analysis for the embedded Trefftz discontinuous…
In this paper, we develop an efficient spectral-Galerkin-type search extension method (SGSEM) for finding multiple solutions to semilinear elliptic boundary value problems. This method constructs effective initial data for multiple…
In the context of isogeometric analysis, we consider two discretization approaches that make the resulting stiffness matrix nonsymmetric even if the differential operator is self-adjoint. These are the collocation method and the…
Acoustic scattering by vehicle surfaces can have significant effects on overall noise levels. In this paper, we present a space-time Galerkin time-domain boundary element method (TDBEM) that offers several distinct advantages over…
In many applications of practical interest, solutions of partial differential equation models arise as critical points of an underlying (energy) functional. If such solutions are saddle points, rather than being maxima or minima, then the…
We study preconditioning techniques for discontinuous Galerkin discretizations of isotropic linear elasticity problems in primal (displacement) formulation. We propose subspace correction methods based on a splitting of the vector valued…
For linear transport and radiative heat transfer equations with random inputs, we develop new generalized polynomial chaos based Asymptotic-Preserving stochastic Galerkin schemes that allow efficient computation for the problems that…
The high-order accurate continuous Galerkin finite element method offers attractive computational efficiency for computational fluid dynamics. A challenge is however spurious oscillations which result for convection dominated flows over…
Discontinuous Galerkin (DG) methods are promising high order discretizations for unsteady compressible flows. Here, we focus on Numerical Weather Prediction (NWP). These flows are characterized by a fine resolution in $z$-direction and low…
The weak Galerkin (WG) finite element method is an effective and flexible general numerical technique for solving partial differential equations. It is a natural extension of the classic conforming finite element method for discontinuous…
We consider the construction of locally conservative fluxes by means of a simple post-processing technique obtained from the finite element solutions of advection diffusion equations. It is known that a naive calculation of fluxes from…
In this paper, we discuss the application of the Generalized Finite Element Method (GFEM) to approximate the solutions of quasilinear elliptic equations with multiple interfaces in one dimensional space. The problem is characterized by…
Over the last few years there have been dramatic advances in our understanding of mathematical and computational models of complex systems in the presence of uncertainty. This has led to a growth in the area of uncertainty quantification as…
This paper investigates the following question: given a Galerkin matrix corresponding to a finite-element discretisation of either the Helmholtz or time-harmonic Maxwell equations with variable coefficients, suppose that the coefficients of…
In this work, we analyze an unfitted discontinuous Galerkin discretization for the numerical solution of the Stokes system based on equal higher-order discontinuous velocities and pressures. This approach combines the best from both worlds,…
In this paper, the authors constructed an auxiliary space multigrid preconditioner for the weak Galerkin finite element method for second-order diffusion equations, discretized on simplicial 2D or 3D meshes. The idea of the auxiliary space…
In this paper we consider second order elliptic partial differential equations with highly varying (heterogeneous) coefficients on a two-dimensional region. The problems are discretized by a composite finite element (FE) and discontinuous…
We deduce a procedure to apply balanced truncation to parameter-dependent differential-algebraic systems. For that we solve multiple projected Lyapunov equations for different parameter values to compute the Gramians that are required for…