arXiv subjects

Guosheng Fu

Publications and source records attributed to Guosheng Fu.

50 records · Page 3Linked to original sources

arXiv2018

Discrete $H^1$-inequalities for spaces admitting M-decompositions

We find new discrete $H^1$- and Poincaré-Friedrichs inequalities by studying the invertibility of the DG approximation of the flux for local spaces admitting M-decompositions. We then show how to use these inequalities to define and analyze new, superconvergent HDG and mixed methods for which the stabilization function is defined in such a way that the approximations satisfy new $H^1$-stability results with which their error analysis is greatly simplified. We apply this approach to define a wide class of energy-bounded, superconvergent HDG and mixed methods for the incompressible Navier-Stokes equations defined on unstructured meshes using, in 2D, general polygonal elements and, in 3D, general, flat-faced tetrahedral, prismatic, pyramidal and hexahedral elements.

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arXiv2018

Exact smooth piecewise polynomial sequences on Alfeld splits

We develop exact piecewise polynomial sequences on Alfeld splits in any spatial dimension and any polynomial degree. An Alfeld split of a simplex is obtained by connecting the vertices of an $n$-simplex with its barycenter. We show that, on these triangulations, the kernel of the exterior derivative has enhanced smoothness. Byproducts of this theory include characterizations of discrete divergence-free subspaces for the Stokes problem, commutative projections, and simple formulas for the dimensions of smooth polynomial spaces.

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arXiv2018

Dispersive behavior of an energy-conserving discontinuous Galerkin method for the one-way wave equation

The dispersive behavior of the recently proposed energy-conserving discontinuous Galerkin (DG) method by Fu and Shu [10] is analyzed and compared with the classical centered and upwinding DG schemes. It is shown that the new scheme gives a significant improvement over the classical centered and upwinding DG schemes in terms of dispersion error. Numerical results are presented to support the theoretical findings.

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arXiv2018

An energy-conserving ultra-weak discontinuous Galerkin method for the generalized Korteweg-De Vries equation

We propose an energy-conserving ultra-weak discontinuous Galerkin (DG) method for the generalized Korteweg-De Vries(KdV) equation in one dimension. Optimal a priori error estimate of order $k + 1$ is obtained for the semi-discrete scheme for the KdV equation without convection term on general nonuniform meshes when polynomials of degree $k\ge 2$ is used. We also numerically observed optimal convergence of the method for the KdV equation with linear or nonlinear convection terms. It is numerically observed for the new method to have a superior performance for long-time simulations over existing DG methods.

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arXiv2018

A high-order HDG method for the Biot's consolidation model

We propose a novel high-order HDG method for the Biot's consolidation model in poroelasticity. We present optimal error analysis for both the semi-discrete and full-discrete (combined with temporal backward differentiation formula) schemes. Numerical tests are provided to demonstrate the performance of the method.

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arXiv2018

Bernstein-Bezier Bases for Tetrahedral Finite Elements

We present a new set of basis functions for H(curl)-conforming, H(div)-conforming, and L2 -conforming finite elements of arbitrary order on a tetrahedron. The basis functions are expressed in terms of Bernstein polynomials and augment the natural H1 -conforming Bernstein basis. The basis functions respect the differential operators, namely, the gradients of the high-order H1 -conforming Bernstein-Bezier basis functions form part of the H(curl)-conforming basis, and the curl of the high-order, non-gradients H(curl)-conforming basis functions form part of the H(div)-conforming basis, and the divergence of the high-order, non-curl H(div)-conforming basis functions form part of the L2-conforming basis. Procedures are given for the efficient computation of the mass and stiffness matrices with these basis functions without using quadrature rules for (piece-wise) constant coefficients on affine tetrahedra. Numerical results are presented to illustrate the use of the basis to approximate representative problems.

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arXiv2018

Optimal energy-conserving discontinuous Galerkin methods for linear symmetric hyperbolic systems

We propose energy-conserving discontinuous Galerkin (DG) methods for symmetric linear hyperbolic systems on general unstructured meshes. Optimal a priori error estimates of order $k+1$ are obtained for the semi-discrete scheme in one dimension, and in multi-dimensions on Cartesian meshes when tensor-product polynomials of degree $k$ are used. A high-order energy-conserving Lax-Wendroff time discretization is also presented. Extensive numerical results in one dimension, and two dimensions on both rectangular and triangular meshes are presented to support the theoretical findings and to assess the new methods. One particular method (with the doubling of unknowns) is found to be optimally convergent on triangular meshes for all the examples considered in this paper. The method is also compared with the classical (dissipative) upwinding DG method and (conservative) DG method with a central flux. It is numerically observed for the new method to have a superior performance for long-time simulations.

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arXiv2017

Parameter-free superconvergent $H(\mathrm{div})$-conforming HDG methods for the Brinkman equations

In this paper, we present new parameter-free superconvergent H(div)-conforming HDG methods for the Brinkman equations on both simplicial and rectangular meshes. The methods are based on a velocity gradient-velocity-pressure formulation, which can be considered as a natural extension of the H(div)-conforming HDG method (defined on simplicial meshes) for the Stokes flow [Math. Comp. 83(2014), pp. 1571-1598]. We obtain optimal error estimates in $L^2$-norms for all the variables in both the Stokes-dominated regime (high viscosity/permeability ratio) and Darcy-dominated regime (low viscosity/permeability ratio). We also obtain superconvergent L^2-estimate of one order higher for a suitable projection of the velocity error, which is typical for (hybrid) mixed methods for elliptic problems. Moreover, thanks to H(div)-conformity of the velocity, our velocity error estimates are independent of the pressure regularity. Preliminary numerical results on both triangular and rectangular meshes in two-space dimensions confirm our theoretical predictions.

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arXiv2017

Fully computable a posteriori error bounds for hybridizable discontinuous Galerkin finite element approximations

We derive a posteriori error estimates for the hybridizable discontinuous Galerkin (HDG) methods, including both the primal and mixed formulations, for the approximation of a linear second-order elliptic problem on conforming simplicial meshes in two and three dimensions. We obtain fully computable, constant free, a posteriori error bounds on the broken energy seminorm and the HDG energy (semi)norm of the error. The estimators are also shown to provide local lower bounds for the HDG energy (semi)norm of the error up to a constant and a higher-order data oscillation term. For the primal HDG methods and mixed HDG methods with an appropriate choice of stabilization parameter, the estimators are also shown to provide a lower bound for the broken energy seminorm of the error up to a constant and a higher-order data oscillation term. Numerical examples are given illustrating the theoretical results.

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arXiv2017

A lowest-order composite finite element exact sequence on pyramids

Composite basis functions for pyramidal elements on the spaces $H^1(Ω)$, $H(\mathrm{curl},Ω)$, $H(\mathrm{div},Ω)$ and $L^2(Ω)$ are presented. In particular, we construct the lowest-order composite pyramidal elements and show that they respect the de Rham diagram, i.e. we have an exact sequence and satisfy the commuting property. Moreover, the finite elements are fully compatible with the standard finite elements for the lowest-order Raviart-Thomas-Nédélec sequence on tetrahedral and hexahedral elements. That is to say, the new elements have the same degrees of freedom on the shared interface with the neighbouring hexahedral or tetrahedra elements, and the basis functions are conforming in the sense that they maintain the required level of continuity (full, tangential component, normal component, ...) across the interface. Furthermore, we study the approximation properties of the spaces as an initial partition consisting of tetrahedra, hexahedra and pyramid elements is successively subdivided and show that the spaces result in the same (optimal) order of approximation in terms of the mesh size $h$ as one would obtain using purely hexahedral or purely tetrahedral partitions.

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arXiv2017

Devising superconvergent HDG methods with symmetric approximate stresses for linear elasticity by $M$-decompositions

We propose a new tool, which we call $M$-decompositions, for devising superconvergent hybridizable discontinuous Galerkin (HDG) methods and hybridized-mixed methods for linear elasticity with strongly symmetric approximate stresses on unstructured polygonal/polyhedral meshes. We show that for an HDG method, when its local approximation space admits an $M$-decomposition, optimal convergence of the approximate stress and superconvergence of an element-by-element postprocessing of the displacement field are obtained. The resulting methods are locking-free. Moreover, we explicitly construct approximation spaces that admit $M$-decompositions on general polygonal elements. We display numerical results on triangular meshes validating our theoretical findings.

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arXiv2016

A systematic construction of finite element commuting exact sequences

We present a systematic construction of finite element exact sequences with a commuting diagram for the de Rham complex in one-, two- and three-space dimensions. We apply the construction in two-space dimensions to rediscover two families of exact sequences for triangles and three for squares, and to uncover one new family of exact sequence for squares and two new families of exact sequences for general polygonal elements. We apply the construction in three-space dimensions to rediscover two families of exact sequences for tetrahedra, three for cubes, and one for prisms; and to uncover four new families of exact sequences for pyramids, three for prisms, and one for cubes.

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arXiv2014

First order least squares method with weakly imposed boundary condition for convection dominated diffusion problems

We present and analyze a first order least squares method for convection dominated diffusion problems, which provides robust L2 a priori error estimate for the scalar variable even if the given data f in L2 space. The novel theoretical approach is to rewrite the method in the framework of discontinuous Petrov - Galerkin (DPG) method, and then show numerical stability by using a key equation discovered by J. Gopalakrishnan and W. Qiu [Math. Comp. 83(2014), pp. 537-552]. This new approach gives an alternative way to do numerical analysis for least squares methods for a large class of differential equations. We also show that the condition number of the global matrix is independent of the diffusion coefficient. A key feature of the method is that there is no stabilization parameter chosen empirically. In addition, Dirichlet boundary condition is weakly imposed. Numerical experiments verify our theoretical results and, in particular, show our way of weakly imposing Dirichlet boundary condition is essential to the design of least squares methods - numerical solutions on subdomains away from interior layers or boundary layers have remarkable accuracy even on coarse meshes, which are unstructured quasi-uniform.

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arXiv2014

An analysis of HDG methods for convection dominated diffusion problems

In this paper, we establish an a priori error analysis of HDG methods with two types of stabilization parameter applied to convection dominated diffusion problem. We show that, using polynomials of degree no greater than k, L2 error of the scalar variable of HDG methods converges in order k + 1/2 if the meshes are not aligned with the flux. This result implies that the HDG approximation converges as fast as conventional DG methods. In the case that the meshes are aligned with the flux, we show that HDG methods with the second type of stabilization parameter achieves optimal rate of convergence. We also discuss preferred form for implementation of HDG methods. In appendix, we show that the spectral condition number of stiffness matrix of HDG methods is independent of the diffusion coefficient. Numerical experiments are presented to verify our theoretical results.

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