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The symmetric C0 interior penalty method is one of the most popular discontinuous Galerkin methods for the biharmonic equation. This paper introduces an automatic local selection of the involved stability parameter in terms of the geometry of the underlying triangulation for arbitrary polynomial degrees. The proposed choice ensures a stable discretization with guaranteed discrete ellipticity constant. Numerical evidence for uniform and adaptive mesh refinement and various polynomial degrees supports the reliability and efficiency of the local parameter selection and recommends this in practice. The approach is documented in 2D for triangles, but the methodology behind can be generalized to higher dimensions, to non-uniform polynomial degrees, and to rectangular discretizations. An appendix presents the realization of our proposed parameter selection in various established finite element software packages. [ABSTRACT FROM AUTHOR]

This paper introduces an extension of the Morley element for approximating solutions to biharmonic equations. Traditionally limited to piecewise quadratic polynomials on triangular elements, the extension leverages weak Galerkin finite element methods to accommodate higher degrees of polynomials and the flexibility of general polytopal elements. By utilizing the Schur complement of the weak Galerkin method, the extension allows for fewest local degrees of freedom while maintaining sufficient accuracy and stability for the numerical solutions. The numerical scheme incorporates locally constructed weak tangential derivatives and weak second order partial derivatives, resulting in an accurate approximation of the biharmonic equation. Optimal order error estimates in both a discrete H 2 norm and the usual L 2 norm are established to assess the accuracy of the numerical approximation. Additionally, numerical results are presented to validate the developed theory and demonstrate the effectiveness of the proposed extension. [ABSTRACT FROM AUTHOR]

In a conforming discontinuous Galerkin (CDG) finite element method, discontinuous P k polynomials are employed. To connect discontinuous functions, the inter-element traces, { u h } and { ∇ u h } , are usually defined as some averages in discontinuous Galerkin finite element methods. But in this CDG finite element method, they are defined as projections of a lifted P k + 4 polynomial from four P k polynomials on neighboring triangles. With properly chosen weak Hessian spaces, when tested by discontinuous polynomials, the variation form can have no inter-element integral, neither any stabilizer. That is, the bilinear form is the same as that of conforming finite elements for solving the biharmonic equation. Such a conforming discontinuous Galerkin finite element method converges four orders above the optimal order, i.e., the P k solution has an O (h k + 5) convergence in L 2 -norm, and an O (h k + 3) convergence in H 2 -norm. A local post-process is defined, which lifts the P k solution to a P k + 4 quasi-optimal solution. Numerical tests are provided, confirming the theory. [ABSTRACT FROM AUTHOR]

Natural vibrations of a rectangular plate with two clamped and two freely supported edges are under consideration. The Bubnov — Galerkin method is used to calculate the first eigenvalues, and the first eigenvalue in the case of a single trial function is calculated with an error smaller than 1%. The results are compared with known data, and eigenforms are given. [ABSTRACT FROM AUTHOR]

We develop an a posteriori error bound for the interior penalty discontinuous Galerkin approximation of the biharmonic equation with continuous finite elements. The error bound is based on the two-energies principle and requires the computation of an equilibrated moment tensor. The natural space for the moment tensor is that of symmetric tensor fields with continuous normal-normal components, and is well-known from the Hellan-Herrmann-Johnson mixed formulation. We propose a construction that is totally local. The procedure can also be applied to the original Hellan–Herrmann–Johnson formulation, which directly provides an equilibrated moment tensor. [ABSTRACT FROM AUTHOR]

We consider an a posteriori error estimator for the Interior Penalty Discontinuous Galerkin (IPDG) approximation of the biharmonic equation based on the Hellan-Herrmann-Johnson (HHJ) mixed formulation. The error estimator is derived from a two-energies principle for the HHJ formulation and amounts to the construction of an equilibrated moment tensor which is done by local interpolation. The reliability estimate is a direct consequence of the two-energies principle and does not involve generic constants. The efficiency of the estimator follows by showing that it can be bounded from above by a residual-type estimator known to be efficient. A documentation of numerical results illustrates the performance of the estimator. [ABSTRACT FROM AUTHOR]

We present and analyze an interior penalty discontinuous Galerkin Isogeometric Analysis (dG-IgA) method for the biharmonic equation in computational domain in R d with d = 2 , 3 . The computational domain consists of several non-overlapping sub-domains or patches. We construct B-Spline approximation spaces which are discontinuous across patch interfaces. We present a priori error estimate in a discrete norm and numerical experiments to confirm the theory. [ABSTRACT FROM AUTHOR]

We present an approximately C 1 -smooth multi-patch spline construction which can be used in isogeometric analysis (IGA). The construction extends the one presented in [1] for two-patch domains. A key property of IGA is that it is simple to achieve high order smoothness within a single patch. However, to represent more complex geometries one often uses a multi-patch construction. In this case, the global continuity for the basis functions is in general only C 0. Therefore, to obtain C 1 -smooth isogeometric functions, a special construction for the basis is needed. Such spaces are of interest when solving numerically fourth-order problems, such as the biharmonic equation or Kirchhoff–Love plate/shell formulations, using an isogeometric Galerkin method. Isogeometric spaces that are globally C 1 over multi-patch domains can be constructed as in [2–6]. The constructions require geometry parametrizations that satisfy certain constraints along the interfaces, so-called analysis-suitable G 1 parametrizations. To allow C 1 spaces over more general multi-patch parametrizations, one needs to increase the polynomial degree and/or to relax the C 1 conditions. Thus, we define function spaces that are not exactly C 1 but only approximately. We adopt the construction for two-patch domains, as developed in [1] , and extend it to more general multi-patch domains. We employ the construction for a biharmonic model problem and compare the results with Nitsche's method. We compare both methods over complex multi-patch domains with non-trivial interfaces. The numerical tests indicate that the proposed construction converges optimally under h -refinement, comparable to the solution using Nitsche's method. In contrast to weakly imposing coupling conditions, the approximate C 1 construction is explicit and no additional terms need to be introduced to stabilize the method/penalize the jump of the derivative at the interface. Thus, the new proposed method can be used more easily as no parameters need to be estimated. [ABSTRACT FROM AUTHOR]

We prove the convergence of an innovative Chebyshev–Gauss–Lobatto (CGL) pseudospectral method applied to fourth order boundary value problems. The proposed method enjoyed all the advantages of pseudospectral methods. Moreover, we can select (N − 3) interior CGL collocation points to enforce the equation, meanwhile we use the remaining four collocation points to assure the boundary conditions. [ABSTRACT FROM AUTHOR]

For the biharmonic problem, we study the convergence of adaptive C0-Interior Penalty Discontinuous Galerkin (C0-IPDG) methods of any polynomial order. We note that C0-IPDG methods for fourth order elliptic boundary value problems have been suggested in [9, 17], whereas residual-type a posteriori error estimators for C0-IPDG methods applied to the biharmonic equation have been developed and analyzed in [8, 18]. Following the convergence analysis of adaptive IPDG methods for second order elliptic problems [6], we prove a contraction property for a weighted sum of the C0-IPDG energy norm of the global discretization error and the estimator. The proof of the contraction property is based on the reliability of the estimator, a quasi-orthogonality result, and an estimator reduction property. Numerical results are given that illustrate the performance of the adaptive C0-IPDG approach. [ABSTRACT FROM AUTHOR]

Two fully discrete, discontinuous Galerkin schemes with time-dynamic, locally refined meshes in space are developed for a fourth-order Cahn-Hilliard equation with an added nonlinear reaction term, a phenomenological model that can describe cancerous tumour growth. The proposed schemes, which are both second-order in time, are based on a primitive-variable discontinuous Galerkin spatial formulation that is valid in any number of space dimensions. We prove that the schemes are convergent, with optimalorder error bounds, even in the case where the mesh is changing with time, provided that the number of mesh changes is bounded by some constant. The schemes represent flexible, high-order accurate alternatives to the standard mixed C0 finite element methods and nonconforming (plate) finite element methods for solving fourth-order parabolic partial differential equations. We conclude the paper with tests showing the convergence of the scheme at the predicted rates and the flexibility of the method for approximating complex solution dynamics efficiently. [ABSTRACT FROM AUTHOR]

SUMMARY A goal-oriented algorithm is developed and applied for hp-adaptive approximations given by the discontinuous Galerkin finite element method for the biharmonic equation. The methodology is based on the dual problem associated with the target functional. We consider three error estimators and analyse their properties as basic tools for the design of the hp-adaptive algorithm. To improve adaptation, the combination of two different error estimators is used, each one at its best efficiency, to guide the tasks of where and how to adapt the approximation spaces. The performance of the resulting hp-adaptive schemes is illustrated by numerical experiments for two benchmark problems. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: In this work, we compare different mesh moving techniques for monolithically-coupled fluid-structure interactions in arbitrary Lagrangian–Eulerian coordinates. The mesh movement is realized by solving an additional partial differential equation of harmonic, linear-elastic, or biharmonic type. We examine an implementation of time discretization that is designed with finite differences. Spatial discretization is based on a Galerkin finite element method. To solve the resulting discrete nonlinear systems, a Newton method with exact Jacobian matrix is used. Our results show that the biharmonic model produces the smoothest meshes but has increased computational cost compared to the other two approaches. [Copyright &y& Elsevier]

We provide an overview of matrix decomposition algorithms (MDAs) for the solution of systems of linear equations arising when various discretization techniques are applied in the numerical solution of certain separable elliptic boundary value problems in the unit square. An MDA is a direct method which reduces the algebraic problem to one of solving a set of independent one-dimensional problems which are generally banded, block tridiagonal, or almost block diagonal. Often, fast Fourier transforms (FFTs) can be employed in an MDA with a resulting computational cost of O( N log N) on an N × N uniform partition of the unit square. To formulate MDAs, we require knowledge of the eigenvalues and eigenvectors of matrices arising in corresponding two-point boundary value problems in one space dimension. In many important cases, these eigensystems are known explicitly, while in others, they must be computed. The first MDAs were formulated almost fifty years ago, for finite difference methods. Herein, we discuss more recent developments in the formulation and application of MDAs in spline collocation, finite element Galerkin and spectral methods, and the method of fundamental solutions. For ease of exposition, we focus primarily on the Dirichlet problem for Poisson's equation in the unit square, sketch extensions to other boundary conditions and to more involved elliptic problems, including the biharmonic Dirichlet problem, and report extensions to three dimensional problems in a cube. MDAs have also been used extensively as preconditioners in iterative methods for solving linear systems arising from discretizations of non-separable boundary value problems. [ABSTRACT FROM AUTHOR]

In this paper, we develop and study approximately smooth basis constructions for isogeometric analysis over two-patch domains. One key element of isogeometric analysis is that it allows high order smoothness within one patch. However, for representing complex geometries, a multi-patch construction is needed. In this case, a C 0 -smooth basis is easy to obtain, whereas C 1 -smooth isogeometric functions require a special construction. Such spaces are of interest when solving numerically fourth-order PDE problems, such as the biharmonic equation and the Kirchhoff–Love plate or shell formulation, using an isogeometric Galerkin method. With the construction of so-called analysis-suitable G 1 (in short, AS- G 1) parametrizations, as introduced in Collin et al. (2016), it is possible to construct C 1 isogeometric spaces which possess optimal approximation properties, cf. (Kapl et al., 2019). These geometries need to satisfy certain constraints along the interfaces and additionally require that the regularity r and degree p of the underlying spline space satisfy 1 ≤ r ≤ p − 2. The problem is that most complex geometries are not AS- G 1 geometries. Therefore, we define basis functions for isogeometric spaces by enforcing approximate C 1 conditions following the basis construction from Kapl et al. (2017). For this reason, the defined function spaces are not exactly C 1 but only approximately. We study the convergence behavior and define function spaces that converge optimally under h -refinement, by locally introducing functions of higher polynomial degree and lower regularity. The convergence rate is optimal in several numerical tests performed on domains with non-trivial interfaces. While an extension to more general multi-patch domains is possible, we restrict ourselves to the two-patch case and focus on the construction over a single interface. • Constructing approximate C 1 smooth bases for isogeometric two-patch domains. • Proof of convergence rates of the jump of the normal derivative across the interface. • Optimal convergence of the isogeometric discretization under h -refinement verifying the results on several numerical examples. [ABSTRACT FROM AUTHOR]

We construct hp-version interior penalty discontinuous Galerkin finite element methods (DGFEMs) for the biharmonic equation, including symmetric and nonsymmetric interior penalty discontinuous Galerkin methods and their combinations: semisymmetric methods. Our main concern is to establish the stability and to develop the a priori error analysis of these methods. We establish error bounds that are optimal in h and slightly suboptimal in p. The theoretical results are confirmed by numerical experiments. [Copyright &y& Elsevier]