Low-order Discontinuous Galerkin Research Articles

Space-time formulation, discretization, and computational performance studies for phase-field fracture optimal control problems

The purpose of this work is the development of space-time discretization schemes for phase-field optimal control problems. Specifically in the optimal control minimization problem, a tracking-type cost functional is minimized to steer the crack via the phase-field variable into a desired pattern. To achieve such optimal solutions, Neumann type boundary conditions need to be determined. First, a time discretization of the forward problem is derived using a discontinuous Galerkin formulation. Here, a challenge is to include regularization terms and the crack irreversibility constraint. The optimal control setting is formulated by means of the Lagrangian approach from which the primal part, adjoint, tangent and adjoint Hessian are derived. Herein the overall Newton algorithm is based on a reduced approach by eliminating the state constraint, namely the displacement and phase-field unknowns, but keeping the control variable as the only unknown. From the low-order discontinuous Galerkin discretization, adjoint time-stepping schemes are finally obtained. Both our formulation and algorithmic developments are substantiated and illustrated with six numerical experiments.

Numerical Analysis of Two Galerkin Discretizations with Graded Temporal Grids for Fractional Evolution Equations

Two numerical methods with graded temporal grids are analyzed for fractional evolution equations. One is a low-order discontinuous Galerkin (DG) discretization in the case of fractional order $$0<\alpha <1$$ , and the other one is a low-order Petrov Galerkin (PG) discretization in the case of fractional order $$1<\alpha <2$$ . By a new duality technique, pointwise-in-time error estimates of first-order and $$ (3-\alpha ) $$ -order temporal accuracies are respectively derived for DG and PG, under reasonable regularity assumptions on the initial value. Numerical experiments are performed to verify the theoretical results.

Hybrid Large Eddy Simulation for low-order Discontinuous Galerkin methods using an explicit filter

In this paper we present a simple, easily implemented and effective approach for explicitly-filtered Large Eddy Simulation with a Discontinuous Galerkin (DG) discretisation for velocity. DG formulations are often desirable due to their stability and increased accuracy, however this can come at greater computational expense due to the additional degrees of freedom in the velocity field. Additionally, data output can also be an issue, due to the increased storage requirements. Here we present a hybrid approach, based upon the construction of an approximation of the velocity shear tensor using information from a projected Continuous Galerkin (CG) version of the discontinuous velocity field. The resulting turbulence algorithm is implemented within Fluidity, an open-source computational fluid dynamics solver. The model is then validated with a well known test case, and shown to agree favourably with published results. Comparisons are also made between the CG/DG hybrid LES with DG-only LES, which demonstrate the superior computational performance of the hybrid model.

Baroclinic waves on the β plane using low-order Discontinuous Galerkin discretisation

A classic Discontinuous Galerkin Method with low-order polynomials and explicit as well as semi-implicit time-stepping is applied to an atmospheric model employing the Euler equations on the β plane. The method, which was initially proposed without regard for the source terms and their balance with the pressure gradient that dominates atmospheric dynamics, needs to be adapted to be able to keep the combined geostrophic and hydrostatic balance in three spatial dimensions. This is achieved inside the discretisation through a polynomial mapping of both source and flux terms without imposing filters between time steps. After introduction and verification of this balancing, the realistic development of barotropic and baroclinic waves in the model is demonstrated, including the formation of a retrograde Rossby wave pattern. A prerequesite is the numerical solution of the thermal wind equation to construct geostrophically balanced initial states in z coordinates with arbitrary prescribed zonal wind profile, offering a new set of test cases for atmospheric models employing z coordinates. The resulting simulations demonstrate that the balanced low-order Discontinuous Galerkin discretisation with polynomial degrees down to k=1 can be a viable option for atmospheric modelling.

Application of low-order Discontinuous Galerkin methods to the analysis of viscoelastic flows

The performance of two low-order discretization schemes in combination with the Discontinuous Galerkin method for the analysis of viscoelastic flows is investigated. An (extended) linear interpolation of the velocity-pressure variables is used in combination with a piecewise discontinuous constant and linear approximation of the extra stresses. Galerkin-leastsquares methodology is applied to stabilize the velocity-pressure discretization. As test problems, the falling sphere in a tube and the stick-slip configuration are studied. The constant stress triangular element converges to high Deborah numbers for a wide variety of material parameters of the Phan-Thien-Tanner model. In particular, for the upper convected Maxwell model, the falling sphere problem converges at least up to Deborah number of 4, while the stick-slip problem converges up to a Deborah number of 25.5.