We study discontinuous Galerkin approximations of the $p$--biharmonic equation from a variational perspective. We propose a discrete variational formulation of the problem based on a appropriate definition of a finite element Hessian and study convergence of the method (without rates) using a weak lower semicontinuity argument.
We present numerical experiments aimed at testing the robustness of the method. We also note a superconvergence effect for some values of $p$.
The generalized weak Galerkin (gWG) finite element method is proposed and analyzed for the biharmonic equation. A new generalized discrete weak second order partial derivative is introduced in the gWG scheme to allow arbitrary combinations of piecewise polynomial functions defined in the interior and on the boundary of general polygonal or polyhedral elements. The error estimates are established for the numerical approximation in a discrete H^2 norm and a L^2 norm. The numerical results are reported to demonstrate the accuracy and flexibility of our proposed gWG method for the biharmonic equat…
This paper proposes and analyzes a fully discrete scheme that discretizes space with an ultra-weak local discontinuous Galerkin scheme and time with the Crank--Nicolson method for the nonlinear biharmonic Schrödinger equation. We first rewrite the problem into a system with a second-order spatial derivative and then apply the ultra-weak discontinuous Galerkin method to the system. The proposed scheme is more computationally efficient compared with the local discontinuous Galerkin method because of fewer auxiliary variables, and unconditionally stable without any penalty terms; it also preserve…
In this paper, we present a study on the effect of penalty term in the mixed Discontinuous Galerkin Finite Element Method for the biharmonic equation proposed by \cite{gudi2008mixed}. The proposed mixed Discontinuous Galerkin Method showed sub-optimal rates of convergence for piecewise quadratic elements and no significant convergence rates for piecewise linear elements. We show that by choosing the penalty term proportional to $|e_k|^{-1}$ instead of $|e_k|^{-3}$, ensures an optimal rate of convergence for the approximation, including for piecewise linear elements. Finally, we present numeric…
We present the analysis of interior penalty discontinuous Galerkin Isogeometric Analysis (dGIGA) for the biharmonic problem on orientable surfaces $Ω\subset \mathbb{R}^3.$ Here, we consider a surface consisting of several non-overlapping patches as typical in multipatch dGIGA. Due to the non-overlapping nature of the patches, we construct NURBS approximation spaces which are discontinuous across the patch interfaces via a penalty scheme. By an appropriate discrete norm, we present \textit{a priori} error estimates for the non-symmetric, symmetric and semi-symmetric interior penalty methods. Fi…
A new stabilizer free weak Galerkin (WG) method is introduced and analyzed for the biharmonic equation. Stabilizing/penalty terms are often necessary in the finite element formulations with discontinuous approximations to ensure the stability of the methods. Removal of stabilizers will simplify finite element formulations and reduce programming complexity. This stabilizer free WG method has an ultra simple formulation and can work on general partitions with polygons/polyhedra. Optimal order error estimates in a discrete $H^2$ for $k\ge 2$ and in $L^2$ norm for $k>2$ are established for the cor…
A Discontinuous Galerkin Method by Patch Reconstruction for Biharmonic Problem
Peking University（北京大学）；State Key Laboratory of Scientific and Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算国家重点实验室）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；Wuhan University（武汉大学）
We develop a basic convergence analysis for an adaptive $\textsf{C}^0\textsf{IPG}$ method for the Biharmonic problem, which provides convergence without rates for all practically relevant marking strategies and all penalty parameters assuring coercivity of the method. The analysis hinges on embedding properties of (broken) Sobolev and BV spaces, and the construction of a suitable limit space. In contrast to the convergence result of adaptive discontinuous Galerkin methods for elliptic PDEs, by Kreuzer and Georgoulis [Math. Comp. 87 (2018), no.~314, 2611--2640], here we have to deal with the fa…
We introduce a residual-based a posteriori error estimator for a novel $hp$-version interior penalty discontinuous Galerkin method for the biharmonic problem in two and three dimensions. We prove that the error estimate provides an upper bound and a local lower bound on the error, and that the lower bound is robust to the local mesh size but not the local polynomial degree. The suboptimality in terms of the polynomial degree is fully explicit and grows at most algebraically. Our analysis does not require the existence of a $\mathcal{C}^1$-conforming piecewise polynomial space and is instead ba…
In this article, we present and analyze a stabilizer-free $C^0$ weak Galerkin (SF-C0WG) method for solving the biharmonic problem. The SF-C0WG method is formulated in terms of cell unknowns which are $C^0$ continuous piecewise polynomials of degree $k+2$ with $k\geq 0$ and in terms of face unknowns which are discontinuous piecewise polynomials of degree $k+1$. The formulation of this SF-C0WG method is without the stabilized or penalty term and is as simple as the $C^1$ conforming finite element scheme of the biharmonic problem. Optimal order error estimates in a discrete $H^2$-like norm and th…
A C^0-Weak Galerkin Finite Element Method for the Biharmonic Equation
Michigan State University（密歇根州立大学）；National Science Foundation（美国国家科学基金会）；University of Arkansas at Little Rock（阿肯色大学小石城分校）；University of Delaware（特拉华大学）
We devise and analyze two hybrid high-order (HHO) methods for the numerical approximation of the biharmonic problem. The methods support polyhedral meshes, rely on the primal formulation of the problem, and deliver $O(h^{k+1})$ $H^2$-error estimates when using polynomials of order $k\ge0$ to approximate the normal derivative on the mesh (inter)faces. Both HHO methods hinge on a stabilization in the spirit of Lehrenfeld--Schöberl for second-order PDEs. The cell unknowns are polynomials of order $(k+2)$ that can be eliminated locally by means of static condensation. The face unknowns approximati…
We present a class of Arbitrary Lagrangian-Eulerian hybridizable discontinuous Galerkin methods for the incompressible flow with moving boundaries and interfaces including two-phase flow with surface tension.
The popular (piecewise) quadratic schemes for the biharmonic equation based on triangles are the nonconforming Morley finite element, the discontinuous Galerkin, the $C^0$ interior penalty, and the WOPSIP schemes. Those methods are modified in their right-hand side $F\in H^{-2}(Ω)$ replaced by $F\circ (JI_{\rm M}) $ and then are quasi-optimal in their respective discrete norms. The smoother $JI_{\rm M}$ is defined for a piecewise smooth input function by a (generalized) Morley interpolation $I_{\rm M}$ followed by a companion operator $J$. An abstract framework for the error analysis in the en…
A new hybrid mixed discontinuous Galerkin finite element (HMDGFE) method is constructed for incompressible miscible displacement problem. In this method, the hybrid mixed finite element (HMFE) procedure is considered to solve pressure and velocity equations, and a new hybrid mixed discontinuous Galerkin procedure is constructed to solve the concentration equation with upwind technique. Compared with other traditional discontinuous Galerkin methods, the new method can reach global systems with less unknowns and sparser stencils. The consistency and conservation of the method are analyzed, the s…
Existing a priori convergence results of the discontinuous Petrov-Galerkin method to solve the problem of linear elasticity are improved. Using duality arguments, we show that higher convergence rates for the displacement can be obtained. Post-processing techniques are introduced in order to prove superconvergence and numerical experiments {\color{black} confirm} our theory.
A discontinuous Galerkin pressure correction numerical method for solving the incompressible Navier-Stokes equations is formulated and analyzed. We prove unconditional stability of the propose scheme. Convergence of the discrete velocity is established by deriving a priori error estimates. Numerical results verify the convergence rates.
In this paper, discontinuous Galerkin finite element methods are applied to one dimensional Rosenau equation. Theoretical results including consistency, a priori bounds and optimal error estimates are established for both semidiscrete and fully discrete schemes. Numerical experiments are performed to validate the theoretical results. The decay estimates are verified numerically for the Rosenau equation.
A conforming discontinuous Galerkin finite element method is introduced for solving the biharmonic equation. This method, by its name, uses discontinuous approximations and keeps simple formulation of the conforming finite element method at the same time. The ultra simple formulation of the method will reduce programming complexity in practice. Optimal order error estimates in a discrete $H^2$ norm is established for the corresponding finite element solutions. Error estimates in the $L^2$ norm are also derived with a sub-optimal order of convergence for the lowest order element and an optimal …
In this paper we propose a novel regularization strategy for the local discontinuous Galerkin method to solve the Hamilton-Jacobi equation in the context of level-set reinitialization. The novel regularization idea works in analogy to shock-capturing schemes for discontinuous Galerkin methods, which are based on finite volume sub-cells. In this spirit, the local discontinuous Galerkin method is combined with an upwind/downwind finite volume sub-cell discretization, which is applied in areas of low regularity. To ensure the applicability on unstructured meshes, the finite volume discretization …
We present a geometric multigrid solver for the Compact Discontinuous Galerkin method through building a hierarchy of coarser meshes using a simple agglomeration method which handles arbitrary element shapes and dimensions. The method is easily extendable to other discontinuous Galerkin discretizations, including the Local DG method and the Interior Penalty method. We demonstrate excellent solver performance for Poisson's equation, provided a flux formulation is used for the operator coarsening and a suitable switch function chosen for the numerical fluxes.
In this paper, we provide a framework of designing the local discontinuous Galerkin scheme for integral fractional Laplacian $(-Δ)^{s}$ with $s\in(0,1)$ in two dimensions. We theoretically prove and numerically verify the numerical stability and convergence of the scheme with the convergence rate no worse than $\mathcal{O}(h^{k+\frac{1}{2}})$.
This paper is devoted to the numerical analysis of a piecewise constant discontinuous Galerkin method for time fractional subdiffusion problems. The regularity of weak solution is firstly established by using variational approach and Mittag-Leffler function. Then several optimal error estimates are derived with low regularity data. Finally, numerical experiments are conducted to verify the theoretical results.
The pressure correction scheme is combined with interior penalty discontinuous Galerkin method to solve the time-dependent Navier-Stokes equations. Optimal error estimates are derived for the velocity in the L$^2$ norm in time and in space. Error bounds for the discrete time derivative of the velocity and for the pressure are also established.
The hybrid Raviart-Thomas mixed discontinuous Galerkin (HRTMDG) method is proposed for solving the Helmholtz equation. With a new energy norm, we establish the existence and uniqueness of the HRTMDG method, and give its convergence analysis. The corresponding error estimate shows that the HRTMDG method has an optimal $L^2$-norm convergence accuracy which is independent of wavenumber.
We devise and analyze a class of interior penalty discontinuous Galerkin methods for nonlinear and nonsmooth variational problems. Discrete duality relations are derived that lead to optimal error estimates in the case of total-variation regularized minimization or obstacle problems. The analysis provides explicit estimates that precisely determine the role of stabilization parameters. Numerical experiments suppport the optimality of the estimates.
We generalize the energy-based discontinuous Galerkin method proposed in [SIAM J. Num. Anal., 53(6):2705-2726, 2015.] to second-order semilinear wave equations. A stability and convergence analysis is presented along with numerical experiments demonstrating optimal convergence for certain choices of the interelement fluxes. Applications to the sine-Gordon equation include simulations of breathers, kink, and anti-kink solitons.
For the stationary advection-diffusion problem the standard continuous Galerkin method is unstable without some additional control on the mesh or method. The interior penalty discontinuous Galerkin method is stable but at the expense of an increased number of degrees of freedom. The hybrid method proposed in [5] combines the computational complexity of the continuous method with the stability of the discontinuous method without a significant increase in degrees of freedom. We discuss the implementation of this method using the finite element library deal.ii and present some numerical experimen…
We present a strongly conservative and pressure-robust hybridizable discontinuous Galerkin method for the coupled time-dependent Navier-Stokes and Darcy problem. We show existence and uniqueness of a solution and present an optimal a priori error analysis for the fully discrete problem when using Backward Euler time stepping. The theoretical results are verified by numerical examples.
