Related papers: Compatible finite element methods for numerical we…
A novel finite element method for the approximation of Maxwell's equations over hybrid two-dimensional grids is studied. The choice of appropriate basis functions and numerical quadrature leads to diagonal mass matrices which allow for…
We present useful connections between the finite difference and the finite element methods for a model boundary value problem. We start from the observation that, in the finite element context, the interpolant of the solution in one…
Adaptive finite elements are the method of choice for accurate simulations of optical components. However as shown recently by Bienstman et al. many finite element mode solvers fail to compute the propagation constant's imaginary part of a…
In a previous article [J. Comp. Phys. $\mathbf{357}$ (2018) 282-304], the mixed mimetic spectral element method was used to solve the rotating shallow water equations in an idealized geometry. Here the method is extended to a smoothly…
This paper is concerned with fully discrete finite element methods for approximating variational solutions of nonlinear stochastic elastic wave equations with multiplicative noise. A detailed analysis of the properties of the weak solution…
This paper develops a smoothing-based postprocessing method for superconvergence in finite element methods. The method applies a few smoothing iterations, such as damped Jacobi, Gauss-Seidel, or conjugate gradient, with initial guess being…
We focus here on a class of fourth-order parabolic equations that can be written as a system of second-order equations by introducing an auxiliary variable. We design a novel second-order fully discrete mixed finite element method to…
The finite element method (FEM) is applied to obtain numerical solutions to a recently derived nonlinear equation for the shallow water wave problem. A weak formulation and the Petrov-Galerkin method are used. It is shown that the FEM gives…
The convergence and optimality of adaptive mixed finite element methods for the Poisson equation are established in this paper. The main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure…
This paper presents a family of spatial discretisations of the nonlinear rotating shallow-water equations that conserve both energy and potential enstrophy. These are based on two-dimensional mixed finite element methods and hence, unlike…
We present new discretisation of the moist compressible Euler equations, using the compatible finite element framework identified in Cotter and Shipton (2012). The discretisation strategy is described and details of the parametrisations of…
The moist shallow water equations offer a promising route for advancing understanding of the coupling of physical parametrisations and dynamics in numerical atmospheric models, an issue known as 'physics-dynamics coupling'. Without moist…
The convergence of an adaptive mixed finite element method for general second order linear elliptic problems defined on simply connected bounded polygonal domains is analyzed in this paper. The main difficulties in the analysis are posed by…
In this paper, we review and systematically compare three finite element discretization methods for a harmonic map heat flow problem from the unit disk in $\mathbb{R}^2$ to the unit sphere in $\mathbb{R}^3$ in an unified framework.…
We construct a finite element method (FEM) for the infinity Laplacian. Solutions of this problem may be singular, which has prompted us to conduct an a posteriori analysis of the method deriving residual based estimators to drive an…
In this paper, we define new unfitted finite element methods for numerically approximating the solution of surface partial differential equations using bulk finite elements. The key idea is that the $n$-dimensional hypersurface, $\Gamma…
Finite element methods provide accurate and efficient methods for the numerical solution of partial differential equations by means of restricting variational problems to finite-dimensional approximating spaces. However, they do not…
The convergence analysis for least-squares finite element methods led to various adaptive mesh-refinement strategies: Collective marking algorithms driven by the built-in a posteriori error estimator or an alternative explicit…
We survey finite element methods for approximating the time harmonic Maxwell equations. We concentrate on comparing error estimates for problems with spatially varying coefficients. For the conforming edge finite element methods, such…
We develop the theory of mixed finite elements in terms of special inverse systems of complexes of differential forms, defined over cellular complexes. Inclusion of cells corresponds to pullback of forms. The theory covers for instance…