A Deluxe FETI-DP Method for Full DG Discretization of Elliptic Problems

This work presents a review and analysis of modern asymptotic methods for analysis of singularly perturbed problems with interior and boundary layers. The central part of the work is a review of the work of the author and his colleagues and disciples. It highlights boundary and initial-boundary value problems for nonlinear elliptic and parabolic partial differential equations, as well as periodic parabolic problems, which are widely used in applications and are called reaction–diffusion and reaction–diffusion–advection equations. These problems can be interpreted as models in chemical kinetics, synergetics, astrophysics, biology, and other fields. The solutions of these problems often have both narrow boundary regions of rapid change and inner layers of various types (contrasting structures, moving interior layers: fronts), which leads to the need to develop new asymptotic methods in order to study them both formally and rigorously. A general scheme for a rigorous study of contrast structures in singularly perturbed problems for partial differential equations, based on the use of the asymptotic method of differential inequalities, is presented and illustrated on relevant problems. The main achievements of this line of studies of partial differential equations are reflected, and the key directions of its development are indicated.

In the paper, we investigate the basic transmission problems arising in the model of fluid-solid acoustic interaction when a piezo-ceramic elastic body ( Ω + ) is embedded in an unbounded fluid domain ( Ω − ). The corresponding physical process is described by boundary-transmission problems for second order partial differential equations. In particular, in the bounded domain Ω + , we have 4 × 4 dimensional matrix strongly elliptic second order partial differential equation, while in the unbounded complement domain Ω − , we have a scalar Helmholtz equation describing acoustic wave propagation. The physical kinematic and dynamic relations mathematically are described by appropriate boundary and transmission conditions. With the help of the potential method and theory of pseudodifferential equations, the uniqueness and existence theorems are proved in Sobolev–Slobodetskii spaces. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we introduce the operators related to mixed boundary value problems for general linear elliptic partial complex differential equations in the unit disc of the complex plane. The solvability of the relevant boundary value problems will be studied by transforming them into singular integral equations.

The nonlocal boundary-value problem for a third order partial differential equation in a Hilbert space H with a self-adjoint positive definite operator A is considered. A stable three-step difference scheme for the approximate solution of the problem is presented. The main theorem on stability of this difference scheme is established. In applications, the stability estimates for the solution of difference schemes of the approximate solution of three nonlocal boundary value problems for third order partial differential equations are obtained. Numerical results for oneand two-dimensional third order partial differential equations are provided.

The purpose of this study is to develop a numerical method for bifurcation analysis of nonlinear partial differential equations, based on the reduction of partial differential equations to ordinary ones, using the Kolmogorov-Arnold theorem. Methods. This paper describes a method for reducing partial differential equations to ordinary ones using the Kolmogorov-Arnold theorem, as well as methods for the bifurcation analysis of nonlinear boundary value problems for ordinary differential equations. Results. The paper presents a new method for solving and bifurcation analysis of nonlinear boundary value problems for partial differential equations, which allow variational formulation. The method was applied to a nonlinear two-dimensional Bratu problem with Dirichlettype boundary conditions. Conclusion. A new method of bifurcation analysis for nonlinear partial differential equations has been developed. Specifically, a method has been proposed for reducing partial different equations to ordinary equations, which allows the use of the developed apparatus of bifurcation analysis for boundary value problems of ordinary differential equations. This method allows conducting bifurcation analysis for arbitrary nonlinear partial differential equations.

On existence and uniqueness of solutions of boundary value problems of fourth order elliptic partial differential equations with variable coefficients

A one-stage meshless method is devised for solving Cauchy boundary value problems of elliptic partial differential equations (PDEs) with variable coefficients. The main idea is to approximate an unknown solution using a linear combination of fundamental solutions and radial basis functions. Compared with the two-stage method of particular solution, the proposed method can deal with more general elliptic PDEs with variable coefficients. Several numerical results in both two- and three-dimensional space show that our proposed method is accurate and effective.

This study focuses on nonlocal boundary value problems for elliptic ordinary and partial differential-operator equations of arbitrary order, defined in Banach-valued function spaces. The region considered here has a varying bound and depends on a certain parameter. Several conditions are obtained that guarantee the maximal regularity and Fredholmness, estimates for the resolvent, and the completeness of the root elements of differential operators generated by the corresponding boundary value problems in Banach-valued weighted spaces. These results are applied to nonlocal boundary value problems for regular elliptic partial differential equations and systems of anisotropic partial differential equations on cylindrical domain to obtain the algebraic conditions that guarantee the same properties.

The nonlocal boundary value problems for regular degenerate differential-operator equations with the parameter are studied. The principal parts of the appropriate generated differential operators are non-self-adjoint. Several conditions for the maximal regularity uniformly with respect to the parameter and the Fredholmness in Banach-valuedLp−spaces of these problems are given. In applications, the nonlocal boundary value problems for degenerate elliptic partial differential equations and for systems of elliptic equations with parameters on cylindrical domain are studied.

In this article boundary value problems for partial differential equations of mixed elliptic–parabolic type are considered. To ensure that the considered problems possess a unique solution, the usual variational existence proof for parabolic problems is extended to the mixed situation. Further, the convergence of approximations computed by a time-space Galerkin method to the solution of the mixed problem is proven and error estimates are given.

On the variational iteration method and other iterative techniques for nonlinear differential equations

We solve a boundary value problem for a first-order partial differential equation in a rectangular domain with a fractional discretely distributed differentiation operator. The fractional differentiation is given by Dzhrbashyan–Nersesyan operators. We construct a representation of the solution and prove existence and uniqueness theorems. The results remain valid for the corresponding equations with Riemann–Liouville and Caputo derivatives. In terms of parameters defining the fractional differential operator, we derive necessary and sufficient conditions for the solvability of the problem.

Surface representation as a boundary-valued problem of partial differential equations (PDE) is an important topic of computer graphics and computer aided design. In existing references, various free-from surfaces were created with a fourth order PDE which is only able to meet the tangential conditions at the surface boundaries. The need for a sixth order PDE in surface modeling arises in two situations: one is to generate surfaces with curvature continuity; and the other is to use curvature values as a user handle for surface shape manipulation. In this chapter, we introduce such a sixth order PDE for free-form surface generation and develop a finite difference method to solve this PDE. We also investigate the effects of boundary curvature and the vector-valued shape parameters on the surface shape.

Existence and uniqueness theorems for boundary value problems for partial differential equations in function spaces

In this study, we investigate the application of variational methods to establish the existence of weak solutions for boundary value problems (BVPs) associated with nonlinear elliptic partial differential equations (PDEs). By employing the direct method in the calculus of variations, we reformulate the given PDEs as minimisation problems over suitable Sobolev spaces, where weak solutions correspond to critical points of appropriately defined energy functionals. The analytical framework incorporates foundational tools such as Sobolev embeddings, compactness theorems, and properties of coercivity and weak lower semi-continuity to ensure the well-posedness of the variational formulation. We present and prove an existence theorem under well-defined conditions on the nonlinear coefficients of the equations, specifically those involving the Carathéodory property, polynomial growth bounds, monotonicity, and symmetry. These conditions are shown to satisfy the hypotheses of the Fucik-Kufner theorem, which guarantees the existence - and, under convexity assumptions, uniqueness - of weak solutions. To demonstrate the utility of the theoretical results, we apply the theorem to selectednonlinear elliptic problems and verify the fulfilment of all required criteria for the existence of solutions. Through this approach, we provide a rigorous and general framework for analysing nonlinear elliptic BVPs without resorting to classical linearization techniques. The findings confirm that variational methods offer powerful, flexible tools for addressing existence questions in complex PDE systems, with relevance to mathematical modelling in physics, engineering, and applied sciences.