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This article introduces a unified mixed finite element framework based on a saddle-point formulation that applies to time-dependent fourth order linear and nonlinear problems with clamped, simply supported, and Cahn-Hilliard type boundary conditions. The classical mixed formulations lead to large matrix systems that demand huge storage and computational time making the schemes expensive, especially for the time-dependent problems. The proposed scheme circumvents this by employing biorthogonal basis functions that lead to sparse and positive-definite systems. The article discusses a mixed finite element method for the biharmonic problem and the time-dependent linear and nonlinear versions of the extended Fisher-Kolmogorov equations equipped with the aforementioned boundary conditions. The wellposedness of the scheme is discussed and a priori error estimates are presented for the semi-discrete and fully discrete finite element schemes. The numerical experiments validate the theoretical estimates derived in the paper. [ABSTRACT FROM AUTHOR]

We consider the solution to the biharmonic equation in mixed form discretized by the Hybrid High-Order (HHO) methods. The two resulting second-order elliptic problems can be decoupled via the introduction of a new unknown, corresponding to the boundary value of the solution of the first Laplacian problem. This technique yields a global linear problem that can be solved iteratively via a Krylov-type method. More precisely, at each iteration of the scheme, two second-order elliptic problems have to be solved, and a normal derivative on the boundary has to be computed. In this work, we specialize this scheme for the HHO discretization. To this aim, an explicit technique to compute the discrete normal derivative of an HHO solution of a Laplacian problem is proposed. Moreover, we show that the resulting discrete scheme is well-posed. Finally, a new preconditioner is designed to speed up the convergence of the Krylov method. Numerical experiments assessing the performance of the proposed iterative algorithm on both two- and three-dimensional test cases are presented. [ABSTRACT FROM AUTHOR]

In this paper, stability analysis and error estimates of a fully discrete local discontinuous Galerkin method for solving the linear bi-harmonic equation are carried out, where the time discretization is treated by a strong-stability-preserving implicit Runge-Kutta scheme. Based on the generalized alternating numerical fluxes, the relationship between the numerical solution and the inner product of the auxiliary solution is established, which plays a key role in obtaining the unconditional stability of the fully discrete local discontinuous Galerkin methods. By carefully introducing reference functions and generalized Gauss-Radau projection, the optimal error estimates are obtained. Numerical experiments are given to demonstrate the stability and accuracy of the fully discrete scheme for the linear bi-harmonic equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear beam equation with the p (x) -biharmonic operator, where the exponent p (x) is a given function. We transform the problem into a system of two differential equations and prove the existence, uniqueness and regularity of the weak solution. Then we formulate the discrete problem associated with that system using the finite element method and prove the existence, uniqueness and stability of the discrete solution. We also investigate the order of convergence and prove some error estimates. Next, we apply the Lagrange basis to obtain an algebraic system of equations. Finally, we implement the computational codes in Matlab software considering the one and two dimensional cases and we present examples to illustrate the theory. • Study of a nonlinear beam equation with the p (x) -biharmonic operator. • Proof of the existence, uniqueness and regularity of the weak solution. • Discrete formulation with the mixed finite element method. • Demonstration of the existence, uniqueness and regularity of the discrete solution. • Numerical simulations with Matlab software for the one and two dimensional cases. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a new mixed method proposed by Rafetseder and Zulehner for Kirchhoff plates and apply it to fourth order eigenvalue problems. Using two auxiliary variables this new mixed method makes it possible to require only H 1 regularity for the displacement and the auxiliary variables, without the demand of a convex domain. We provide a direct comparison, specifically in view of convergence orders, to the C 0 -IPG method and Ciarlet-Raviart's mixed method of vibration problems with the boundary conditions of the clamped plate and the simply supported plate. The numerical experiments are done with the open-source finite element library deal.II and include the implementation of the coupling of finite elements with elements on the boundary to incorporate non-trivial boundary conditions. [ABSTRACT FROM AUTHOR]

Fourth-order differential equations play an important role in many applications in science and engineering. In this paper, we present a three-field mixed finite-element formulation for fourth-order problems, with a focus on the effective treatment of the different boundary conditions that arise naturally in a variational formulation. Our formulation is based on introducing the gradient of the solution as an explicit variable, constrained using a Lagrange multiplier. The essential boundary conditions are enforced weakly, using Nitsche's method where required. As a result, the problem is rewritten as a saddle-point system, requiring analysis of the resulting finite-element discretization and the construction of optimal linear solvers. Here, we discuss the analysis of the well-posedness and accuracy of the finite-element formulation. Moreover, we develop monolithic multigrid solvers for the resulting linear systems. Two and three-dimensional numerical results are presented to demonstrate the accuracy of the discretization and efficiency of the multigrid solvers proposed. [ABSTRACT FROM AUTHOR]

In this paper, a new strategy to establish less time-consuming upwind compact difference method with adjusted dissipation is introduced for solving the incompressible Navier-Stokes (N-S) equations in the streamfunction-velocity form efficiently. By weighted combination of the numerical solutions calculated using the upwind term and the downwind term of the general third-order upwind compact scheme (UCD3), a new fourth-order compact formulation and a third-order upwind compact formulation with adjusted dissipation nature are proposed for computing the first derivatives. Further, they are used to approximate the biharmonic term and the convective terms in the streamfunction-velocity formulation of the N-S equations, respectively. Meanwhile, the first derivatives of the streamfunction (velocities) in the coefficients in the convective terms are solved by the newly proposed fourth-order compact formulation. Temporal discretization for the streamfunction-velocity formulation is addressed with the help of the second-order Crank-Nicolson scheme. Moreover, the newly proposed scheme for the linear models is proved to be unconditionally stable by virtue of the discrete Fourier analysis. Finally, five numerical problems, viz. the analytic solution, Taylor-Green vortex problem, doubly periodic double shear layer flow problem, lid-driven square cavity flow problem and two-sided square cavity flow problem are solved numerically to verify the efficiency and accuracy of the present method. Results solved by the present method match well with the analytic solutions and the existing results proving the accuracy of it. What is more, it is less time-consuming and has lower dissipation than the existing method [20]. [ABSTRACT FROM AUTHOR]

We consider a linearized partial data Calderón problem for biharmonic operators extending the analogous result for harmonic operators (Dos Santos Ferreira et al., 2009). We construct special solutions and utilize Segal-Bargmann transform to recover lower order perturbations. [ABSTRACT FROM AUTHOR]

This paper proposes a novel framework for simulating the dynamics of beams on elastic foundations. Specifically, partial differential equations modeling Euler–Bernoulli and Timoshenko beams on the Winkler foundation are simulated using a causal physics-informed neural network (PINN) coupled with transfer learning. Conventional PINNs encounter challenges in handling large space–time domains, even for problems with closed-form analytical solutions. A causality-respecting PINN loss function is employed to overcome this limitation, effectively capturing the underlying physics. However, it is observed that the causality-respecting PINN lacks generalizability. We propose using solutions to similar problems instead of training from scratch by employing transfer learning while adhering to causality to accelerate convergence and ensure accurate results across diverse scenarios. The primary contribution of this paper lies in introducing a causality-respecting PINN loss function in the context of structural engineering and coupling it with transfer learning to enhance the generalizability of PINNs in simulating the dynamics of beams on elastic foundations. Numerical experiments on the Euler–Bernoulli beam highlight the efficacy of the proposed approach for various initial conditions, including those with noise in the initial data. Furthermore, the potential of the proposed method is demonstrated for the Timoshenko beam in an extended spatial and temporal domain. Several comparisons suggest that the proposed method accurately captures the inherent dynamics, outperforming the state-of-the-art physics-informed methods under standard L 2 -norm metric and accelerating convergence. [ABSTRACT FROM AUTHOR]

Biharmonic wave equations are of importance to various applications including thin plate analyses. The innovation of this work comes through the numerical approximation of their solutions by a C 1 -conforming in space and time finite element approach. Therein, the smoothness properties of solutions to the continuous evolution problem are embodied. Time discretization is based on a combined Galerkin and collocation technique. For space discretization the Bogner–Fox–Schmit element is applied. Optimal order error estimates are proven. The convergence and performance properties are illustrated by numerical experiments with complex wave profiles in homogeneous and heterogeneous media, illustrating that the approach offers high potential also for sophisticated multi-physics and/or multi-scale systems. [ABSTRACT FROM AUTHOR]

A conforming discontinuous Galerkin (CDG) C 0 - P k finite element method is introduced for solving the biharmonic equation on triangular and tetrahedral meshes. A C 0 - P k finite element function is a continuous and piecewise polynomial of degree k on a triangular or tetrahedral mesh. The CDG method is based on taking weak divergence on the gradient of C 0 - P k finite elements. Optimal order error estimates in both a discrete H 2 norm and the L 2 norm are established. Numerical results are presented to verify the theory. [ABSTRACT FROM AUTHOR]

In this article, we discuss goal-oriented a posteriori error estimation for the biharmonic plate bending problem. The error for a numerical approximation of a goal functional is represented by several computable estimators. One of these estimators is obtained using the dual-weighted residual method, which takes advantage of an equilibrated moment tensor. Then, an abstract unified framework for the goal-oriented a posteriori error estimation is derived based on the equilibrated moment tensor and the potential reconstruction that provides a guaranteed upper bound for the error of a numerical approximation for the goal functional. In particular, C 0 interior penalty and discontinuous Galerkin finite element methods are employed for the practical realisation of the estimators. Numerical experiments are performed to illustrate the effectivity of the estimators. [ABSTRACT FROM AUTHOR]

In this paper, a staggered cell-centered discontinuous Galerkin method is developed for the biharmonic problem with the Steklov boundary condition. Our approach utilizes a first-order system form of the biharmonic problem and can handle fairly general meshes possibly including hanging nodes, which favors adaptive mesh refinement. Optimal order error estimates in L 2 norm can be proved for all the variables. Moreover, the approximation of the primal variable superconverges in L 2 norm to a suitably chosen projection without requiring additional regularity. Residual type error estimators are proposed, which can guide adaptive mesh refinement to deliver optimal convergence rates even for solutions with singularity. Numerical experiments confirm that the optimal convergence rates in L 2 norm can be achieved for all the variables. Moreover, all the provided residual type error estimators show the desired results. In particular, the numerical results demonstrate that the proposed scheme on a polygonal approximation of the disk works well for the classic Babuška example. [ABSTRACT FROM AUTHOR]

The plate bending problems in the Kirchhoff–Love model can be reduced to biharmonic equations. In the previous papers (Kim et al., 2020; Kim et al., 2022), we derived stress intensity factors (SIF) extraction formulas of a biharmonic equation Δ 2 u = f in a non-convex (cracked, L -shaped) domain with various boundary conditions such as clamped, simply connected, free, and mixed. In previous papers, we could not show a general SIF formula related to Poisson's ratio ν contained in the free boundary condition. Moreover, the previous SIF extraction formula obtained for a given Poisson's ratio requires cumbersome line integrals. In this paper, removing the line integrals in the extraction formulas, we show improved SIF extraction formulas applicable to biharmonic equations with free boundary conditions. Thus, the new, improved SIF extraction formulas for free boundary conditions become more straightforward and can be applied to plate bending problems. [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]

The a posteriori error analysis of the classical Argyris finite element methods dates back to 1996, while the optimal convergence rates of associated adaptive finite element schemes are established only very recently in 2021. It took a long time to realize the necessity of an extension of the classical finite element spaces to make them hierarchical. This paper establishes the novel adaptive schemes for the biharmonic eigenvalue problems and provides a mathematical proof of optimal convergence rates towards a simple eigenvalue and numerical evidence thereof. This makes the suggested algorithm highly competitive and clearly justifies the higher computational and implementational costs compared to low-order nonconforming schemes. The numerical experiments provide overwhelming evidence that higher polynomial degrees pay off with higher convergence rates and underline that adaptive mesh-refining is mandatory. Five computational benchmarks display accurate reference eigenvalues. [ABSTRACT FROM AUTHOR]

Physics-Informed Neural Networks (PINNs) have proven effective in solving partial differential equations (PDEs), especially when some data are available by seamlessly blending data and physics. However, extending PINNs to high-dimensional and even high-order PDEs encounters significant challenges due to the computational cost associated with automatic differentiation in the residual loss function calculation. Herein, we address the limitations of PINNs in handling high-dimensional and high-order PDEs by introducing the Hutchinson Trace Estimation (HTE) method. Starting with the second-order high-dimensional PDEs, which are ubiquitous in scientific computing, HTE is applied to transform the calculation of the entire Hessian matrix into a Hessian vector product (HVP). This approach not only alleviates the computational bottleneck via Taylor-mode automatic differentiation but also significantly reduces memory consumption from the Hessian matrix to an HVP's scalar output. We further showcase HTE's convergence to the original PINN loss and its unbiased behavior under specific conditions. Comparisons with the Stochastic Dimension Gradient Descent (SDGD) highlight the distinct advantages of HTE, particularly in scenarios with significant variability and variance among dimensions. We further extend the application of HTE to higher-order and higher-dimensional PDEs, specifically addressing the biharmonic equation. By employing tensor-vector products (TVP), HTE efficiently computes the colossal tensor associated with the fourth-order high-dimensional biharmonic equation, saving memory and enabling rapid computation. The effectiveness of HTE is illustrated through experimental setups, demonstrating comparable convergence rates with SDGD under memory and speed constraints. Additionally, HTE proves valuable in accelerating the Gradient-Enhanced PINN (gPINN) version as well as the Biharmonic equation. Overall, HTE opens up a new capability in scientific machine learning for tackling high-order and high-dimensional PDEs. [ABSTRACT FROM AUTHOR]

Mesh degeneration is a bottleneck for fluid–structure interaction (FSI) simulations and for shape optimization via the method of mappings. In both cases, an appropriate mesh motion technique is required. The choice is typically based on heuristics, e.g., the solution operators of partial differential equations (PDE), such as the Laplace or biharmonic equation. Especially the latter, which shows good numerical performance for large displacements, is expensive. Moreover, from a continuous perspective, choosing the mesh motion technique is to a certain extent arbitrary and has no influence on the physically relevant quantities. Therefore, we consider approaches inspired by machine learning. We present a hybrid PDE-NN approach, where the neural network (NN) serves as parameterization of a coefficient in a second order nonlinear PDE. We ensure existence of solutions for the nonlinear PDE by the choice of the neural network architecture. Moreover, we present an approach where a neural network corrects the harmonic extension such that the boundary displacement is not changed. In order to avoid technical difficulties in coupling finite element and machine learning software, we work with a splitting of the monolithic FSI system into three smaller subsystems. This allows to solve the mesh motion equation in a separate step. We assess the quality of the learned mesh motion technique by applying it to a FSI benchmark problem. In addition, we discuss generalizability and computational cost of the learned mesh motion operators. [ABSTRACT FROM AUTHOR]

We present an isogeometric collocation method for solving the biharmonic equation over planar bilinearly parameterized multi-patch domains. The developed approach is based on the use of the globally C 4 -smooth isogeometric spline space (Kapl and Vitrih, 2021) to approximate the solution of the considered partial differential equation, and proposes as collocation points two different choices, namely on the one hand the Greville points and on the other hand the so-called superconvergent points. Several examples demonstrate the potential of our collocation method for solving the biharmonic equation over planar multi-patch domains, and numerically study the convergence behavior of the two types of collocation points with respect to the L 2 -norm as well as to equivalents of the H s -seminorms for 1 ≤ s ≤ 4. In the studied case of spline degree p = 9 , the numerical results indicate in case of the Greville points a convergence of order O (h p − 3) independent of the considered (semi)norm, and show in case of the superconvergent points an improved convergence of order O (h p − 2) for all (semi)norms except for the equivalent of the H 4 -seminorm, where the order O (h p − 3) is anyway optimal. [ABSTRACT FROM AUTHOR]

This paper focuses on the advancement of weak Galerkin (WG) finite element methods for addressing two-dimensional and three-dimensional Biharmonic interface problems with polygonal/polyhedral meshes. The WG method has been demonstrated to be accurate and efficient, providing optimal order error estimates in discrete H 2 and standard L 2 norms. A series of extensive numerical tests are conducted to validate the WG solutions, showcasing the flexibility, stability, and robustness of the proposed method for handling both smooth and complicated interfaces. [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]

In this paper, we first prove that a quadratic form from S m to S n is non-harmonic biharmonic if and only if it has constant energy density (m + 1) / 2. Then, we give a positive answer to an open problem raised in [1] concerning the structure of non-harmonic biharmonic quadratic forms. As a direct application, using classification results for harmonic quadratic forms, we infer classification results for non-harmonic biharmonic quadratic forms. [ABSTRACT FROM AUTHOR]

We are concerned with multiplicity of solutions for differential inclusions involving the p -biharmonic operator, submitted to a general potential multivalued boundary condition which, among others, covers many particular boundary conditions that are frequently invoked in the literature. Our approach is a variational one and relies on non-smooth critical point theory. [ABSTRACT FROM AUTHOR]

In [13] , we derived stress intensity factors (SIF) extraction formulas of a biharmonic equation Δ 2 u = f in a cracked domain whose crack faces have clamped boundary conditions. In this paper, we extend our investigation to the derivation of the SIF extraction formulas of the biharmonic equation in non-convex polygonal domains containing cracks or reentrant corners when various BC such as clamped (CC), simply supported (SS), free (FF), or mixed conditions (CS, FC, SF) are imposed on the boundary. We prove that the SIF is expressed as the integral of f Ψ s k ⁎ − u Δ 2 Ψ s k ⁎ for a cut-off function Ψ and a dual singular function s k ⁎ on a small neighborhood of the singularity. The dual singular function is determined in this paper. For a numerical approximation of u , we proposed an iteration method as well as a direct finite element method. For a finite element solution of Δ 2 u = f , we have to use C 1 -continuous basis functions. For continuous differentiable basis functions, we use either C 1 -continuous B-spline basis functions or the conventional Hermite basis function. Because of several advantages of B-spline functions in imposing complex boundary conditions, we choose B-spline basis functions for the finite element approximation of u. Moreover, in the direct numerical method, we propose to use implicitly enriched basis functions that resemble the singularities. [ABSTRACT FROM AUTHOR]

In this work, a time-fractional biharmonic equation with a Caputo derivative of fractional order α ∈ (0 , 1) is considered, whose solutions exhibit a weak singularity at initial time t = 0. For this problem, a system of two second-order differential equations is derived by introducing a intermediate variable p = − Δ u , then discretised the system using the standard finite element method in space together with the L1 discretisation of Caputo derivative on graded mesh in time. The H 1 -norm stability result of the method is established, and then a sharp H 1 -norm local convergent result is presented. Finally, numerical experiments are provided to further verify our theoretical analysis for each fixed value of α. [ABSTRACT FROM AUTHOR]

In this paper we present a pseudospectral method in the disk. Unlike the methods already known, the disk is not duplicated. Moreover, we solve the Laplace equation under nonhomogeneous Dirichlet, Neumann and Robin boundary conditions, as well as the biharmonic equation subject to nonhomogeneous Dirichlet conditions, by only using the elements of the corresponding differentiation matrices. It is worth mentioning that we do not use any quadrature, nor need to solve any decoupled system of ordinary differential equations, nor use any pole condition, nor require any lifting. We also solve several numerical examples to show the spectral convergence. The pseudospectral method developed in this paper is applied to estimate Sherwood numbers integrating the mass flux to the disk, and it can be implemented to solve Lotka–Volterra systems and nonlinear diffusion problems involving chemical reactions. [ABSTRACT FROM AUTHOR]

In this paper, the localized Trefftz method (LTM) is proposed to accurately and efficiently solve two-dimensional boundary value problems, governed by Laplace and biharmonic equations, in complex domains. The LTM is formed by combining the classical indirect Trefftz method and the localization approach, so the LTM, free from mesh and numerical quadrature, has great potential for solving large-scale problems. For problems in multiply-connected domains, the solutions expressions in the proposed LTM is much simpler and more compact than that in the conventional indirect Trefftz method due to the localization concept and the overlapping subdomains. In the proposed LTM, both of the interior nodes and boundary nodes are required and the algebraic equation at each node, represents the satisfaction of governing equation or boundary condition, can be derived by implementing the Trefftz method at every subdomain. By enforcing the satisfaction of governing equations at every interior node and of boundary conditions at every boundary node, a sparse system of linear algebraic equations can be yielded. Then, the numerical solution of the proposed LTM can be efficiently obtained by solving the sparse system. Several numerical examples in simply-connected and multiply-connected domains are provided to demonstrate the accuracy and simplicity of the proposed LTM. Besides, the extremely-accurate solutions of the LTM are simultaneously demonstrated. [ABSTRACT FROM AUTHOR]

In this article, we investigate p (x) -biharmonic equations involving Leray–Lions type operators and Hardy potentials. Leray–Lions type operators include p -Laplacian, p -Laplacian-like, p (x) -Laplacian, (p , q) -Laplacian, (p (x) , q (x)) -Laplacian operators and so forth. Moreover, fourth-order Leray–Lions type elliptic equations with variable exponents are seldom mentioned in previous papers. Some new theorems of the existence on the generalized solutions are reestablished for such equations when the Leray–Lions type operator and the nonlinearity satisfy suitable hypotheses in variable exponent Lebesgue spaces. [ABSTRACT FROM AUTHOR]

In order to perform isogeometric analysis with increased smoothness on complex domains, trimming, variational coupling or unstructured spline methods can be used. The latter two classes of methods require a multi-patch segmentation of the domain, and provide continuous bases along patch interfaces. In the context of shell modelling, variational methods are widely used, whereas the application of unstructured spline methods on shell problems is rather scarce. In this paper, we therefore provide a qualitative and a quantitative comparison of a selection of unstructured spline constructions, in particular the D-Patch, Almost- C 1 , Analysis-Suitable G 1 and the Approximate C 1 constructions. Using this comparison, we aim to provide insight into the selection of methods for practical problems, as well as directions for future research. In the qualitative comparison, the properties of each method are evaluated and compared. In the quantitative comparison, a selection of numerical examples is used to highlight different advantages and disadvantages of each method. In the latter, comparison with weak coupling methods such as Nitsche's method or penalty methods is made as well. In brief, it is concluded that the Approximate C 1 and Analysis-Suitable G 1 converge optimally in the analysis of a bi-harmonic problem, without the need of special refinement procedures. Furthermore, these methods provide accurate stress fields. On the other hand, the Almost- C 1 and D-Patch provide relatively easy construction on complex geometries. The Almost- C 1 method does not have limitations on the valence of boundary vertices, unlike the D-Patch, but is only applicable to biquadratic local bases. Following from these conclusions, future research directions are proposed, for example towards making the Approximate C 1 and Analysis-Suitable G 1 applicable to more complex geometries. [ABSTRACT FROM AUTHOR]

A decoupled finite element method for the triharmonic equation in two dimensions is developed in this paper. The triharmonic equation can be decoupled into two biharmonic equations and one symmetric tensor-valued Stokes equation. Two biharmonic equations are discretized by the Morley element. An H 1 (S) -conforming finite element in arbitrary dimension is constructed and applied for solving the tensor-valued Stokes equation. The convergence analysis is presented for the resulting discrete method. Numerical results are provided to verify the theoretical convergence rates. [ABSTRACT FROM AUTHOR]

In this paper, a posteriori error estimates for the Weak Galerkin finite element methods (WG-FEMs) for second order elliptic problems are derived in terms of an H 1 − equivalent energy norm. Corresponding estimators based on the helmholtz decomposition yield globally upper and locally lower bounds for the approximation errors of the WG-FEMs. Especially, the error analysis of our methods is proved to be valid for polygonal meshes (e.g., hybrid, polytopal non-convex meshes and those with hanging nodes) under general assumptions. In addition, the work can make adaptive WG-FEMs solving partial differential equations such as stokes equations and biharmonic equations on polygonal meshes possible. Finally, we verify the theoretical findings by a few numerical examples. [ABSTRACT FROM AUTHOR]

• Violation of the Saint–Venant compatibility condition is considered. • Equation for the Airy stress function in the non-Euclidean model is obtained. • The Airy stress function is calculated through the incompatibility function. • Residual stress are determined in the framework of the non-Euclidean model. We introduced the Airy stress function in the non-Euclidean model of a continuous medium, for which the Saint-Venant compatibility condition for deformations is not satisfied. For this function, an inhomogeneous biharmonic equation is obtained whose right side is determined by the incompatibility of deformations. The internal stresses was shown to be composed of elastic stresses and stresses determined by the incompatibility of deformations. The theoretical results obtained are used to analyse experimental data on the study of residual stresses. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in an inverse problem for the biharmonic equation to recover the Robin coefficients on a non-accessible part of the boundary in simply connected planar domain from a measured Riquier-Neumann data on the accessible part of the boundary. Our work is based on a system of nonlinear and ill-posed integral equations which is solved by Tikhonov regularization method to complete missing Cauchy data. The minimization problem will be considered to solve the inverse problem, and recover the Robin coefficients on a known boundary curve. We show the feasibility of this method by numerical reconstructions. [ABSTRACT FROM AUTHOR]

Within the framework of the Kirchhoff-Love theory, a thin homogeneous layer (called adhesive) of small width between two plates (called adherents) is considered. It is assumed that elastic properties of the adhesive layer depend on its width which is a small parameter of the problem. Our goal is to perform an asymptotic analysis as the parameter goes to zero. It is shown that depending on the softness or stiffness of the adhesive, there are seven distinct types of interface conditions. In all cases, we establish weak convergence of the solutions of the initial problem to the solutions of limiting ones in appropriate Sobolev spaces. The asymptotic analysis is based on variational properties of solutions of corresponding equilibrium problems. [ABSTRACT FROM AUTHOR]

We derive formulas extracting stress intensity factors of the biharmonic equations on cracked domains with clamped (or simply supported or free) boundary conditions along the crack faces. Each of these formulas can be written in terms of the integral of the given source function multiplied by the cut-off dual singularity and the integral of the unknown true solution multiplied by the cut-off dual singularity over the unit disk. The unknown true solution in the extraction formulas is calculated by either the Implicitly Enriched Galerkin Method or the Iteration Methods. The former was developed by the authors (Kim, Oh, Palta, Kim) and the latter proposed in this paper is the sum of the solution of a regular biharmonic equation and singular functions with iteratively estimated stress intensity factors as coefficients. We show the Iteration Methods quickly converge and the proposed Enrichment Method yields highly accurate stress intensity factors. We also demonstrate that for a known true solution, the extraction formulas yield exact stress intensity factors. [ABSTRACT FROM AUTHOR]

There are three criteria when we model with phase-field equations. The first one is the evolution of phase separation. The second one is energy dissipation law. The last one is whether we have the law of conservation of mass or not. If we minimize the Ginzburg–Landau free-energy functional, the kinetics of phase separation inherently follows. However, the other properties such as energy dissipation and mass conservation are more difficult to model. Hitherto, there have been a few proposals to conserve mass using the Allen–Cahn (AC) equation, but we do not have the energy dissipation law in these models when considered in light of local effects of mass-correction. For the reasons, certain researchers still prefer to use the Cahn–Hilliard equation, e.g. biharmonic equation instead of using the Allen–Cahn equation when we consider a mass conserving case. This paper is concerned with numerical solutions of the AC equation in the mass-conserving space, where the solution is corrected at each iteration for mass conservation. We introduce two different methods for solving the conservative AC equation with periodic boundary conditions. In this work, we have the AC equation in common, but use two different mass correction operators. Note that both of mass correction operators satisfy the discrete energy dissipation property. First, we project the solution of the AC equation to the mass conserving space, and then recover the linear convex splitting scheme for the conventional mass-conserving equation. The linear convex splitting scheme is well-known but the convergence of numerical solutions is elaborately studied in terms of gradient-projection. Next, we propose the other energy-minimizing method on the mass-conserving space. It translates the solution of the AC equation to the mass conserving space via an energy-dissipation operator. The dissipation of discrete energy functional for the proposed method is also established. As far as the author knows, no report has been presented for the numerical scheme having discrete energy dissipation property of the mass-conserving AC equation without using the nonlocal and constant multiplier. To verify the common properties of mass-projection and proposed methods numerically, we simulate the evolutions for phase separation, dissipation of energy, and mass conservation. We moreover consider the short and long time stages of evolutions to see the difference between two methods. These several tests are provided for its applications of the proposed method for modeling natural phenomena. [ABSTRACT FROM AUTHOR]

We give some theoretical as well as computational results on Laplace and Maxwell constants, i.e., on the smallest constants c n > 0 arising in estimates of the form | u | L 2 (Ω) ≤ c 0 | grad u | L 2 (Ω) , | E | L 2 (Ω) ≤ c 1 | curl E | L 2 (Ω) , | H | L 2 (Ω) ≤ c 2 | div H | L 2 (Ω) . Besides the classical de Rham complex we investigate the complex of elasticity and the complex related to the biharmonic equation and general relativity as well using the general functional analytical concept of Hilbert complexes. We consider mixed boundary conditions and bounded Lipschitz domains of arbitrary topology. Our numerical aspects are presented by examples for the de Rham complex in 2D and 3D which not only confirm our theoretical findings but also indicate some interesting conjectures. [ABSTRACT FROM AUTHOR]

• We propose a novel stabilized SAV approach for solving the phase field model for diblock copolymers. • The proposed schemes are second-order accurate, provably unconditionally energy stable, non-iterative. • The added linear stabilization term is shown to be crucial enhance the stability while keeping the required accuracy. • One only need to solve three decoupled linear equations at each time step. • We further prove the unconditional energy stabilities rigorously and present numerous 2D and 3D simulations. In this paper, we consider numerical approximations to solving the Cahn-Hilliard phase field model for diblock copolymers. We combine the recently developed SAV (scalar auxiliary variable) approach with the stabilization technique to arrive at a novel stabilized-SAV method, where a crucial linear stabilization term is added to enhancing the stability and keeping the required accuracy while using the large time steps. The scheme is very easy-to-implement and fast in the sense that one only needs to solve two decoupled fourth-order biharmonic equations with constant coefficients at each time step. We further prove the unconditional energy stability of the scheme rigorously. Through the comparisons with some other prevalent schemes like the fully-implicit, convex-splitting, and non-stabilized SAV scheme for some benchmark numerical examples in 2D and 3D, we demonstrate the stability and the accuracy of the developed scheme numerically. [ABSTRACT FROM AUTHOR]

• A novel scheme is developed to simulate the thin plate bending problems. • The proposed method is meshless without numerical integration and iteration. • The presented method is quite accurate. This study develops a simple and effective numerical technique, which aims to accurately and quickly address thin plate bending problems. Based on the given boundary conditions, the thin plate homogenization function is constructed and a family of two-parameter homogenization functions are derived. Then, the superposition of homogenization functions method for the thin plate, the clamped plate, and the simply supported plate is obtained, which is meshless without numerical integration and iteration with the merits of easy-to-program and easy-to-implement. Six numerical experiments are employed to verify the effectiveness, accuracy and convergence of the proposed novel strategy. The proposed method is evaluated by the comparisons with the analytical solutions and the referenced solutions. It can be observed that the proposed method is quite accurate for the thin plate, the clamped plate, and the simply supported plate problems. [ABSTRACT FROM AUTHOR]

We employ a radial basis function (RBF) - differential quadrature (DQ) method for the numerical solution of elliptic boundary value problems in annular domains. With an appropriate selection of collocation points, for any choice of RBF, both the coefficient and right hand side matrices in the systems appearing in this discretization possess block circulant structures. These linear systems can thus be solved efficiently using matrix decomposition algorithms (MDAs) and fast Fourier transforms (FFTs). In particular, we consider problems governed by the Poisson equation, the inhomogeneous biharmonic equation and the inhomogeneous Cauchy–Navier equations of elasticity. In addition to its simplicity, the proposed method can both achieve high accuracy and solve large-scale problems. The feasibility of the proposed techniques is illustrated by several numerical examples. [ABSTRACT FROM AUTHOR]