A biharmonic boundary value problem with a singularity is one of the mathematical models of processes in fracture mechanics. It is necessary to have estimates of the function norms in the neighborhood of the singularity point to study the existence and uniqueness of the Rν-generalized solution, its coercive and differential properties of biharmonic boundary value problems with a corner singularity. This paper establishes estimates of a function in the neighborhood of a singularity point in the norms of weighted Lebesgue spaces through its norms in weighted Sobolev spaces over the entire domain, with a minimum weight exponent. In addition, we obtain an estimate of the function norm in a boundary strip for the degeneration of a function on the entire boundary of the domain. These estimates will be useful not only for studying differential problems with singularity, but also in estimating the convergence rate of an approximate solution to an exact one in the weighted finite element method.

Mathematical models with a singularity play a decisive role in predicting the development of processes in fracture mechanics and physics. In the engineering literature, computational methods are widely studied for problems of electromagnetism, hydrodynamics and the theory of elasticity with a singularity caused by the presence of re-entrant corners on the boundary of a domain. We have constructed a weighted finite element method based on the introduction of the definition of an R_ν -generalized solution for the listed problems . This method allows finding a solution with high accuracy compared to the classical finite element method. At the same time, it retains the simplicity of the stiffness matrix structure, unlike other numerical methods of increased accuracy. To use weighted finite element method effectively, it is necessary to correctly set the control parameters of the approach. In this paper, the weighted finite element method and found the body of optimal parameters for finding an approximate solution to the Lamé system in domains with a boundary containing re-entrant corners in the range from π to 2π are presented.

Boundary value problems for wave vector equations with singularity are used in mathematical models of electromagnetism problems in domains with a boundary containing reentrant corners. The solutions to such problems do not belong to the Sobolev space W^1_2 and the approximate solution by the classical finite element method has a low rate of convergence to the exact solution. We define an R_ν -generalized solution to the boundary value problem for a wave vector equation with corner singularity in a set of the weighted Sobolev-Monk space. A weighted vector finite element method (WVFEM) is constructed to find an approximate R_ν -generalized problem. The WVFEM basis functions contain weighting functions to a degree depending on the sizes of the reentrant corners at the domain boundary. This allows us to weaken the influence of the singularity on the accuracy of finding the solution. The method has a convergence rate of O(h) in the norm of the weighted space L_2,α( )^2 . Numerical calculations of model problems confirmed the theoretical estimate of the convergence rate.

The L-stable second order single diagonal implicit Runge-Kutta method for finding a solution of nonlinear Navier-Stokes equations in rotation form in the L-shaped domain is applied. An Rν-generalized solution of the problem in weighted sets is defined. An unweighted finite element method is constructed to find its approximate solution. The results of numerical experiments have shown the advantage of the proposed method over the classical ones.

A weighted finite element method based on the introduction of the definition of an Rν-generalized solution for the system of Lamé equations is constructed. An estimate for the rate of convergence of the approximate solution by the weighted finite element method to the Rν-generalized solution with the rate of convergence O(h) is proved. A general body of optimal control parameters for WFEM is identified, which allows one to find an approximate solution with high accuracy in domains with a boundary containing re-entrant corners α in the range from π to 2π.

For an approximate solution of the non-stationary nonlinear Navier-Stokes equations for the flow of an incompressible viscous fluid, depending on the set of input data and the geometry of the domain, the area of optimal parameters in the variables $\nu$ and $\nu^{\ast}$ is experimentally determined depending on $\delta$ included in the definition of the $R_{\nu}$-generalized solution of the problem and the degree of the weight function in the basis of the finite element method. To discretize the problem in time, the Runge-Kutta methods of the first and second orders were used. The areas of optimal parameters for various values of the incoming angles are established.

Mathematical models of fracture physics and mechanics are boundary value problems for differential equations and systems of equations with a singularity. There are two classes of problems with a singularity: with coordinated and uncoordinated degeneracy of the input data, depending on the behavior of the coefficients of the equation. Finite element methods with the first order of convergence rate O(h) have been created to find an approximate solution to these problems. We construct a scheme of the weighted finite element method of high degree of accuracy for the boundary value problem with uncoordinated degeneracy of the input data and singularity of the solution. The rate of convergence of an approximate solution of the proposed finite element method to the exact Rν-generalized solution in the weight set W2,ν+β2+21(Ω,δ) is investigated. The estimation of finite element approximation O(h2) is established.

Mathematical models with a singularity play a decisive role in predicting the development of processes in fracture mechanics. A weighted finite element method based on the introduction of the definition of an Rν-generalized solution for the system of Lamé equations with corner singularity is constructed. An estimate for the convergence of an approximate solution by a weighted finite element method to an Rν-generalized solution at a rate of O(h), that is, without loss of accuracy, is proved. For effective use of a weighted finite element method, it is necessary to correctly set the control parameters of the approach for performing calculations. An algorithm for determining the optimal parameters the weighted finite element method for finding an approximate solution to the Lamé system in domains with a boundary containing re-entrant corners ranging from π to 2π is developed. The general body of optimal parameters for weighted finite element method is determined.

In this paper, we introduced the notion of an Rν-generalized solution of the Oseen problem in rotation form in two-dimensional polygonal domain with a reentrant corner at the boundary. Its existence and uniqueness in the nonsymmetric variational formulation of the problem is established. An estimate related to the conservation of the energy balance of the approximation velocity field for the Navier–Stokes problem in the rotation form is obtained. A weighted finite element method is constructed. A series of numerical experiments of test examples based on the proposed method is carried out. A comparative analysis of the approach with the classical FEM is performed. The results of numerical experiments showed a significant advantage of the proposed approach.

The stress–strain state of a pipeline segment with a branch pipe of smaller diameter, which is under internal pressure, is investigated. The mathematical model of this mechanical object is a boundary value problem for a system of six partial differential equations. This boundary value problem describes the deformation of an elastic surface containing a singular line and immersed in three-dimensional space. From this three-dimensional mathematical model in a domain with curved boundary, a resolving boundary value problem in a planar domain is obtained. An algorithm for numerical analysis of the deformed state of a pipeline with a branch pipe by the finite element method is created and implemented. An approach to suppress the error of the numerical solution, which significantly improved the accuracy of the results, is proposed. As a result of numerical experiments, it is found that the proposed algorithms and methods make it possible to approximate a given solution of the reduced pipe deformation problem with a branch pipe by its numerical solution with high accuracy.

The weighted finite element method allows to find an approximate solution to a boundary value problem with a singularity faster in 106 times than the classical finite element method for a given error equal to 10-3. In this case, it is required to apply the necessary control parameters in the weighted finite element method. The body of optimal parameters is determined on the basis of carrying out and analysing a series of numerical experiments. In this paper we propose an algorithm for processing the results of calculations and determining the body of optimal parameters for the Dirichlet problem and the Lame ' system in a domain with one reentrant corner on the boundary taking values from 7r to 27r.

The steady-state nonlinear problem governing the Newtonian flow of an incompressible viscous fluid in convection form in L-shaped domain is considered. The solution as [Formula: see text]-generalized one in weighted sets is determined. Unlike the classical definition of the generalized solution in a weak formulation, which has a symmetric structure, our setting has an asymmetric one so that a special inf-sup condition in weighted sets is proved. A new weighted finite element scheme for an approximate [Formula: see text]-generalized solution is constructed. A converging iterative method to solve the sequence of linear problems with block preconditioning of their matrices is introduced. The results of numerical simulations of several tests have shown an advantage over the standard methods so that an approximate [Formula: see text]-generalized solution tends to the exact one with a rate almost twice in order exceed, relative to the grid step, than classical approaches. The result is achieved without using mesh refined in the neighborhood of a reentrant corner.

A B S T R A C T The resolving equations for mathematical model of the stress-strain state of a curved thin walled cylinder were derived. This model is based on Koiter's-Vlasov's theory of moment shells. A method was proposed for approximate solution of a mathematical model equations on the basis of a sequential asymptotic expansion of unknown functions into a small parameter series and representation of the expansion coefficients in the form of Fourier series. Using this method, a one-dimensional statement of the problem was obtained. Limitations on parameters in the shell equations are indicated for which such problem transformation is possible. For the mathematical model of the curved thin-walled cylinder one-dimensional equations in two different formulations were obtained. Conditions determining applicability limits of the constructed one-dimensional mathematical models were proved. Numerical experiments were carried out and it was found that the constructed one-dimensional mathematical model approximates the original problem with high accuracy. From an applied point of view, curved thin-walled cylinder simulates a pipeline section.(c) 2022 Elsevier Inc. All rights reserved.

This paper discusses the finite element approximation of the Nikol'skij–Lizorkin problem with degeneracy on the entire boundary of the domain. The triangulation of the domain with a special compression of nodes to the boundary of the domain was carried out. It was established that the approximation to the exact solution has first-order convergence in the norm of the Sobolev weighted space W2,α1(Ω) at special exponent of the degree of mesh compression. Numerical experiments confirmed the established estimate of the convergence rate.

Equations of elastic cylindrical shells in term displacements for the T-shaped connection of pipes are derived. Three-dimensional mathematical model is constructed within the framework of the membrane shells theory. Geometric and force conjugation conditions are set on the pipe intersection line, and boundary conditions are imposed on the ends of the pipes. Complete three-dimensional mathematical model is presented in a Cartesian coordinate system, to achieve a unified approach to solving the problem. Dimension of the original problem is reduced by one. This result is obtained from the symmetry conditions of the mechanical system with respect to the plane of the T-shaped connection. The resulting two-dimensional boundary value problem is divided into two sub-problems, each of which is posed in a rectangular domain. Conjugation conditions are eliminated from the final formulation of the boundary value problem. A numerical experiment is carried out, which proved the permissibility of replacing the conjugation conditions with sleeve contact. Numerical examples showed a weak dependence of the approximation of the conjugation conditions by the sleeve contact on the choice of the dimension of the finite elements: three-dimensional or two-dimensional. The existence of a stress field singularity in the vicinity of the shell joining line is established.

The problem that is obtained using implicit time integration of a unsteady nonlinear incompressible Navier-Stokes equations in the rotation form in non-convex polygonal domain is considered. The weighted finite element method based on the concept of an Rν -generalized solution is constructed. The advantage of the proposed approach over classical approximations is numerically established.

The weighted finite element method makes it possible to find an approximate solution of a boundary value problem with corner singularity without loss of accuracy. The construction of this numerical method is based on the introduction of the concept of an Rν-generalized solution for a boundary value problem with a singularity. In this paper, special weighted sets based on the corresponding operators from the definition of the Rν-generalized solution of the Stokes problem in a nonsymmetric variational formulation are introduced. The properties and relationships of these weighted sets are established.

In this manuscript, the Nikol’skii-Lizorkin problem with degeneracy on the entire boundary of the domain is considered. The degeneracy is caused by the behavior of the coefficients and right-hand sides of the problem on the boundary. The triangulation of the domain with a special compression to the boundary of the domain is carried out and the finite element method is constructed. An estimate for the rate of convergence of an approximate solution to an exact solution with the second order in the mesh step in the L2(Ω) space norm is proved. Numerical experiments have confirmed the established estimate of the convergence rate.