Finite Element Approximation of Time-Fractional Fourth-Order Problem with Nonlocal Diffusion: Existence–Uniqueness and Error Bounds

A Petrov-Galerkin method for nonlocal convection-dominated diffusion problems

Nonlinear dynamics of heterogeneous shells Part 1. Statics and dynamics of heterogeneous variable stiffness shells

In this paper, we consider a contact problem between a viscoelastic Bresse beam and a deformable obstacle. The well-known normal compliance contact condition is used to model the contact. The existence of a unique solution to the continuous problem is proved using the Faedo-Galerkin method. An exponential decay property is also obtained defining an adequate Liapunov function. Then, using the finite element method and the implicit Euler scheme, a finite element approximation is introduced. A discrete stability property and a priori error estimates are proved. Finally, some numerical experiments are performed to demonstrate the decay of the discrete energy and the numerical convergence.

In this paper we study the biharmonic equation with Navier boundary conditions in a polygonal domain. In particular, we propose a method that effectively decouples the fourth-order problem as a system of Poisson equations. Our method differs from the naive mixed method that leads to two Poisson problems but only applies to convex domains; our decomposition involves a third Poisson equation to confine the solution in the correct function space, and therefore can be used in both convex and nonconvex domains. A $C^0$ finite element algorithm is in turn proposed to solve the resulting system. In addition, we derive optimal error estimates for the numerical solution on both quasi-uniform meshes and graded meshes. Numerical test results are presented to justify the theoretical findings.

In this paper, we consider a class of discontinuous Galerkin (DG) methods for one-dimensional nonlocal diffusion (ND) problems. The nonlocal models are integral equations that are widely used in describing many physical phenomena with long-range interactions. The ND problem is the nonlocal analog of the classic diffusion problem, and the nonlocality disappears as the interaction radius (horizon) vanishes, then the ND problem recovers the classic diffusion problem. Under certain conditions, the exact solution to the ND problem may contain discontinuities, and this is one of the properties of the ND problem that sets it apart from the classic diffusion problem. It is natural to adopt the DG method to compute the ND problems since the DG method shows its great advantages in resolving problems with discontinuities in the computational fluid dynamics in the past several decades. Based on \cite{DuCAMC2020}, we construct the DG methods with different penalty terms, and the proposed DG methods have their local counterparts as the horizon vanishes. This indicates the proposed methods will converge to the existing DG schemes as the horizon vanishes, which is crucial to obtain the {\it asymptotic compatibility}. Rigorous proofs are provided to demonstrate the stability, error estimates, and asymptotic compatibility of the proposed DG schemes. To see the nonlocal diffusion effect, we also consider the time-dependent convection-diffusion problems with nonlocal diffusion. We conduct several numerical experiments, including accuracy tests and Burgers' equation with nonlocal diffusion, and various horizons are taken to show the good performance of the proposed algorithm and validate the theoretical findings.

In this paper, we get the existence of two positive solutions for a fourth-order problem with Navier boundary condition. Our nonlinearity has a critical growth, and the method is a local minimum theorem obtained by Bonanno.

In this paper, we consider the unsteady flow of a compressible micropolar fluid between two moving, thermally isolated parallel plates. The fluid is characterized as viscous and thermally conductive, with polytropic thermodynamic properties. Although the mathematical model is inherently three-dimensional, we assume that the variables depend on only a single spatial dimension, reducing the problem to a one-dimensional formulation. The non-homogeneous boundary conditions representing the movement of the plates lead to moving domain boundaries. The model is formulated in mass Lagrangian coordinates, which leads to a time-invariant domain. This work focuses on numerical simulations of the fluid flow for different configurations. Two computational approaches are used and compared. The first is based on the finite difference method and the second is based on the Faedo–Galerkin method. To apply the Faedo–Galerkin method, the boundary conditions must first be homogenized and the model equations reformulated. On the other hand, in the finite difference method, the non-homogeneous boundary conditions are implemented directly, which reduces the computational complexity of the numerical scheme. In the performed numerical experiments, it was observed that, for the same accuracy, the Faedo–Galerkin method was approximately 40 times more computationally expensive compared to the finite difference method. However, on a dense numerical grid, the finite difference method required a very small time step, which could lead to an accumulation of round-off errors. On the other hand, the Faedo–Galerkin method showed the convergence of the solutions as the number of expansion terms increased, despite the higher computational cost. Comparisons of the obtained results show good agreement between the two approaches, which confirms the consistency and validity of the numerical solutions.

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In this paper, we consider the Petrov–Galerkin spectral method for fourth-order elliptic problems on rectangular domains subject to non-homogeneous Dirichlet boundary conditions. We derive some sharp results on the orthogonal approximations in one and two dimensions, which play important roles in numerical solutions of higher-order problems. By applying these results to a fourth-order problem, we establish the H2-error and L2-error bounds of the Petrov–Galerkin spectral method. Numerical experiments are provided to illustrate the high accuracy of the proposed method and coincide well with the theoretical analysis. Copyright © 2013 John Wiley & Sons, Ltd.

Presented as Invited Lecture at the 10th Conference on the Mathematics of Finite Elements and Applications, Brunel University, June 1999. This paper is devoted to the a priori and a posteriori error analysis of the hp-version of the discontinuous Galerkin finite element method for partial differential equations of hyperbolic and nearly-hyperbolic character. We consider second-order partial differential equations with nonnegative characteristic form, a large class of equations which includes convection-dominated diffusion problems, degenerate elliptic equations and second-order problems of mixed elliptic-hyperbolic-parabolic type. An a priori error bound is derived for the method in the so-called DG-norm which is optimal in terms of the mesh size h; the error bound is either 1 degree or 1/2 degree below optimal in terms of the polynomial degree p, depending on whether the problem is convection-dominated, or diffusion-dominated, respectively. In the case of a first-order hyperbolic equation the error bound is hp-optimal in the DG-norm. For first-order hyperbolic problems, we also discuss the a posteriori error analysis of the method and implement the resulting bounds into an hp-adaptive algorithm. The theoretical findings are illustrated by numerical experiments.

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Problem statement: This research purposely brought up to solve complicated equations such as partial differential equations, integral equations, Integro-Differential Equations (IDE), stochastic equations and others. Many physical phenomena contain mathematical formulations such integro-differential equations which are arise in fluid dynamics, biological models and chemical kinetics. In fact, several formulations and numerical solutions of the linear Fredholm integro-differential equation of second order currently have been proposed. This study presented the numerical solution of the linear Fredholm integro-differential equation of second order discretized by using finite difference and trapezoidal methods. Approach: The linear Fredholm integro-differential equation of second order will be discretized by using finite difference and trapezoidal methods in order to derive an approximation equation. Later this approximation equation will be used to generate a dense linear system and solved by using the Generalized Minimal Residual (GMRES) method. Results: Several numerical experiments were conducted to examine the efficiency of GMRES method for solving linear system generated from the discretization of linear Fredholm integro-differential equation. For the comparison purpose, there are three parameters such as number of iterations, computational time and absolute error will be considered. Based on observation of numerical results, it can be seen that the number of iterations and computational time of GMRES have declined much faster than Gauss-Seidel (GS) method. Conclusion: The efficiency of GMRES based on the proposed discretization is superior as compared to GS iterative method.

Analysis of a multidimensional thermoviscoelastic contact problem under the Green–Lindsay theory

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