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In the research community, there exists the strong belief that a continuous Galerkin scheme is notoriously unstable and additional stabilization terms have to be added to guarantee stability. In the first part of the series [6], the…
A weak Galerkin (WG) finite element method for solving the stationary Stokes equations in two- or three- dimensional spaces by using discontinuous piecewise polynomials is developed and analyzed. The variational form we considered is based…
We present a realizability-preserving numerical method for solving a spectral two-moment model to simulate the transport of massless, neutral particles interacting with a steady background material moving with relativistic velocities. The…
In this paper we propose a new finite element discretization for the two-field formulation of poroelasticity which uses the elastic displacement and the pore pressure as primary variables. The main goal is to develop a numerical method with…
We construct a cut finite element method for the membrane elasticity problem on an embedded mesh using tangential differential calculus. Both free membranes and membranes coupled to 3D elasticity are considered. The discretization comes…
This article devises a new numerical method for first-order transport problems by using the primal-dual weak Galerkin (PD-WG) finite element method recently developed in scientific computing. The PD-WG method is based on a variational…
Stability and error analysis of a hybridized discontinuous Galerkin finite element method for Stokes equations is presented. The method is locally conservative, and for particular choices of spaces the velocity field is point-wise…
Spacetime discontinuous Galerkin (SDG) finite element methods are used to solve such PDEs involving space and time variables arising from wave propagation phenomena in important applications in science and engineering. To support an…
We present a hybrid continuous and discontinuous Galerkin spectral element approximation that leverages the advantages of each approach. The continuous Galerkin approximation is used on interior element faces where the equation properties…
This paper outlines a conservative transport scheme for scalar tracers within a compatible finite element model for geophysical fluid equations. Instead of using the advective transport equation for a mixing ratio, a conservative transport…
We consider a boundary value problem involving a Riemann-Liouville fractional derivative of order $\alpha\in (3/2,2)$ on the unit interval $(0,1)$. The standard Galerkin finite element approximation converges slowly due to the presence of…
We present a space-time continuous-Galerkin finite element method for solving incompressible Navier-Stokes equations. To ensure stability of the discrete variational problem, we apply ideas from the variational multi-scale method. The…
We design a novel provably stable discontinuous Galerkin spectral element (DGSEM) approximation to solve systems of conservation laws on moving domains. To incorporate the motion of the domain, we use an arbitrary Lagrangian-Eulerian…
This is a survey on the theory of adaptive finite element methods (AFEMs), which are fundamental in modern computational science and engineering but whose mathematical assessment is a formidable challenge. We present a self-contained and…
Allen--Cahn equation with constant and degenerate mobility, and with polynomial and logarithmic energy functionals is discretized using symmetric interior penalty discontinuous Galerkin (SIPG) finite elements in space. We show that the…
The numerical solution of strain gradient-dependent continuum problems has been dogged by continuity demands on the basis functions. For most commonly accepted models, solutions using the finite element method demand $C^{1}$ continuity of…
Estimating the travel time of ultrasound in an inhomogeneous medium is crucial for high-quality imaging, as with an accurate estimate of the distribution of speed of sound, phase aberration, which is normally viewed as one of the reasons…
Analysis of an interface stabilised finite element method for the scalar advection-diffusion-reaction equation is presented. The method inherits attractive properties of both continuous and discontinuous Galerkin methods, namely the same…
A newly developed weak Galerkin method is proposed to solve parabolic equations. This method allows the usage of totally discontinuous functions in approximation space and preserves the energy conservation law. Both continuous and…
The magnetic flow meter is one of the best possible choice for the measurement of flow rate of liquid metals in fast breeder reactors. Due to the associated complexities in the measuring environment, theoretical evaluation of their…