We present an arbitrary order discontinuous Galerkin finite element method for solving the biharmonic interface problem on the unfitted mesh. The approximation space is constructed by a patch reconstruction process with at most one degree freedom per element. The discrete problem is based on the symmetric interior penalty method and the jump conditions are weakly imposed by the Nitsche's technique. The C^2-smooth interface is allowed to intersect elements in a very general fashion and the stability near the interface is naturally ensured by the patch reconstruction. We prove the optimal a priori error estimate under the energy norm and the L^2 norm. Numerical results are provided to verify the theoretical analysis.
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…
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…
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…
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 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…
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 …
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…
In this article, goal-oriented a posteriori error estimation for the biharmonic plate bending problem is considered. The error for approximation of goal functional is represented by an estimator which combines dual-weighted residual method and equilibrated moment tensor. An abstract unified framework for the goal-oriented a posteriori error estimation is derived. In particular, $C^0$ interior penalty and discontinuous Galerkin finite element methods are employed for practical realization. The abstract estimation is based on equilibrated moment tensor and potential reconstruction that provides …
The fully nonconforming virtual element method for biharmonic problems
Politecnico di Milano（米兰理工大学）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；Istituto di Matematica Applicata e Tecnologie Informatiche - CNR（意大利国家研究委员会应用数学与信息技术研究所）
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…
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…
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…
We propose a new least squares finite element method to solve the Stokes problem with two sequential steps. The approximation spaces are constructed by patch reconstruction with one unknown per element. For the first step, we reconstruct an approximation space consisting of piecewise curl-free polynomials with zero trace. By this space, we minimize a least squaresfunctional toobtain thenumericalapproximationstothe gradientof thevelocityand the pressure. In the second step, we minimize another least squares functional to give the solution to the velocity in the reconstructed piecewise divergenc…
We propose a numerical method to solve the Monge-Ampere equation which admits a classical convex solution. The Monge-Ampere equation is reformulated into an equivalent first-order system. We adopt a novel reconstructed discontinuous approximation space which consists of piecewise irrotational polynomials. This space allows us to solve the first order system in two sequential steps. In the first step, we solve a nonlinear system to obtain the approximation to the gradient. A Newton iteration is adopted to handle the nonlinearity in the system. The approximation to the primitive variable is obta…
We devise and analyze $C^0$-conforming hybrid high-order (HHO) methods to approximate biharmonic problems with either clamped or simply supported boundary conditions. $C^0$-conforming HHO methods hinge on cell unknowns which are $C^0$-conforming polynomials of order $(k+2)$ approximating the solution in the mesh cells and on face unknowns which are polynomials of order $k\ge0$ approximating the normal derivative of the solution on the mesh skeleton. Such methods deliver $O(h^{k+1})$ $H^2$-error estimates for smooth solutions. An important novelty in the error analysis is to lower the minimal r…
Preconditioned Reconstructed Discontinuous Approximation For Elliptic Interface Problem on Unfitted Mesh
Peking University（北京大学）；Hangzhou International Innovation Institute of Beihang University（北京航空航天大学杭州创新研究院）；Sichuan University（四川大学）；Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）
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 adapt a symmetric interior penalty discontinuous Galerkin method using a patch reconstructed approximation space to solve elliptic eigenvalue problems, including both second and fourth order problems in 2D and 3D. It is a direct extension of the method recently proposed to solve corresponding boundary value problems, and the optimal error estimates of the approximation to eigenfunctions and eigenvalues are instant consequences from existing results. The method enjoys the advantage that it uses only one degree of freedom on each element to achieve very high order accuracy, which is highly pr…
Regular convergence, together with various other types of convergence, has been studied since the 1970s for the discrete approximations of linear operators. In this paper, we consider the eigenvalue approximation of compact operators whose spectral problem can be written as the eigenvalue problem of some holomophic Fredholm operator function. Focusing on the finite element methods (conforming, discontinuous Galerkin, etc.), we show that the regular convergence of discrete holomorphic operator functions follows from the approximation property of the finite element spaces and the compact converg…
We present the two-dimensional unstructured grids extension of the a posteriori local subcell correction of discontinuous Galerkin (DG) schemes introduced in [52]. The technique is based on the reformulation of DG scheme as a finite volume (FV) like method through the definition of some specific numerical fluxes referred to as reconstructed fluxes. High-order DG numerical scheme combined with this new local subcell correction technique ensures positivity preservation of the solution, along with a low oscillatory and sharp shocks representation. The main idea of this correction procedure is to …
Runge-Kutta Discontinuous Galerkin Method Based on Flux Vector Splitting with Constrained Optimization-based TVB(D)-minmod Limiter for Solving Hyperbolic Conservation Laws
A Hybrid High-Order discretisation of the Brinkman problem robust in the Darcy and Stokes limits
University of Bergamo（贝加莫大学）；Institut Montpelliérain Alexander Grothendieck, Univ Montpellier, CNRS（蒙彼利埃亚历山大格罗滕迪克研究所，蒙彼利埃大学，法国国家科学研究中心）；Monash University（莫纳什大学）
$\boldsymbol{H}(\textbf{curl})$-reconstruction of piecewise polynomial fields with application to $hp$-a posteriori nonconforming error analysis for Maxwell's equations
