Discontinuous Boundary Data Research Articles

An error estimate for finite element approximation to elliptic PDEs with discontinuous Dirichlet boundary data

This note provides an error estimate for finite element approximation to elliptic partial differential equations (PDEs) with discontinuous Dirichlet boundary data. Solutions of problems of this type are not in H1 and, hence, the standard variational formulation is not valid. To circumvent this difficulty, an error estimate of a finite element approximation in the W1,r(Ω) (0<r<2) norm is obtained through a regularization by constructing a continuous approximation of the Dirichlet boundary data.

Inverse problems for a class of elliptic obstacle problems involving multivalued convection term and weighted $(p,q)$-Laplacian

In this paper, we study the inverse problem of estimating discontinuous parameters and boundary data in a nonlinear elliptic obstacle problem involving a nonhomogeneous, nonlinear partial differential operator, which is given as a sum of a weighted p-Laplacian and a weighted q-Laplacian (called the weighted -Laplacian), a multivalued convection term (a multivalued reaction term depending on the gradient), a multivalued boundary condition and an obstacle constraint. Under general hypotheses on the data, we prove the existence of a nontrivial solution to the elliptic obstacle problem, which depends on the first eigenvalue of the Steklov eigenvalue problem for the p-Laplacian. We also establish the boundedness and the closedness of the solution set to the problem. For the parameter-to-solution map, we give a convergence result of the Kuratowski type and prove the solvability of the inverse problem.

Curvilinear integrals of discontinuous functions over nonrectifiable paths and Riemann boundary‐value problem

We study a generalization of the concept of curvilinear integral for the case where path of integration is nonrectifiable, and integrand has discontinuities. Then we apply that integrals for solving of the Riemann boundary‐value problem for domains with nonrectifiable boundaries and discontinuous boundary data. The research is supported by RFBR (grant 18‐31‐00060) and is performed according to a special programme of the Russian government supporting research at Kazan Federal University.Mathematical Methods in the Applied Sciences journal encourages authors to share the data and other artifacts supporting the results in the paper by archiving it in an appropriate public repository. Authors may provide a data availability statement, including a link to the repository they have used, in order that this statement can be published in their paper. Shared data should be cited.

Propagation of boundary-induced discontinuity in stationary radiative transfer and its application to the optical tomography

We consider a boundary value problem of the stationary transport equation with the incoming boundary condition in two or three dimensional bounded convex domains. We discuss discontinuity of the solution to the boundary value problem arising from discontinuous incoming boundary data, which we call the boundary-induced discontinuity. In particular, we give two kinds of sufficient conditions on the incoming boundary data for the boundary-induced discontinuity. We propose a method to reconstruct the attenuation coefficient from jumps in boundary measurements.

(Non)uniqueness of minimizers in the least gradient problem

Minimizers in the least gradient problem with discontinuous boundary data need not be unique. However, all of them have a similar structure of level sets. Here, we give a full characterization of the set of minimizers in terms of any one of them and discuss stability properties of an approximate problem.

A Local Meshless Method for Steady State Convection Dominated Flows

A stable localized meshless method (SLMM) is proposed for convection dominated PDEs exhibiting boundary layer. Some cases of continuous and discontinuous boundary data as well as continuous and discontinuous source function with constant and variable convection coefficients are considered. In this approach, the localized meshless method is implemented on specialized sub-domains embedded with flow direction of underlying fluid. The proposed method is based on flow featured overlapping sub-domains, called stencils. The concept of flow direction is used to construct good quality stencils having the ability to capture flow features, such as boundary layer, accurately. Numerical experiments are presented to compare the proposed method with the finite-difference method on special grid (FDSG), the standard finite-element method, hybridized SUPG method, hybridized upwind method, residual-free bubbles (RFB) method and other meshless methods. Numerical results confirm that the new approach is accurate and efficient for solving a wide class of one-, two-, and three-dimensional convection-dominated PDEs. In some cases, performance of the SLMM is comparable and sometimes better than the mesh-based finite-element and finite-difference methods.

Interaction of fan–jump–fan composite waves in a two-dimensional steady jet for van der Waals gases

We consider a two-dimensional (2D) steady jet generated by gas streaming in parallel supersonic flow out of a duct and into the atmosphere. When the gas is a van der Waals gas, at the corners at the exit of the duct, the parallel supersonic flow expands symmetrically into centered simple waves, jump–fan (JF) composite waves, fan–jump composite waves, or fan–jump–fan (FJF) composite waves. This paper studies the interaction of the symmetric centered FJF composite waves in the jet. The interaction of the FJF composite waves is more involved in comparison to our earlier work on the interaction of the symmetric centered JF composite waves, since the flow in the FJF composite wave interaction zone is rotational. To construct the flow in the interaction zone, we consider a Goursat-type boundary value problem with discontinuous boundary data associated with the 2D and isentropic steady Euler equations. The existence of a global piecewise smooth solution to this Goursat problem is obtained by a new method based on characteristics.

ON THE EXPONENTIAL CONVERGENCE OF THE METHOD OF FUNDAMENTAL SOLUTIONS

It is well known that the method of fundamental solutions (MFS) is a numerical method of exponential convergence. In other words, the logarithmic error is proportional to the node number of spatial discretization. In this study, the exponential convergence of the MFS is demonstrated by solving the Laplace equation in domains of rectangles, ellipses, amoeba-like shapes, and rectangular cuboids. In the solution procedure, the sources of the MFS are located as far as possible and the instability resulted from the ill-conditioning of system matrix is avoided by using the multiple precision floating-point reliable (MPFR) library. The results converge faster for the cases of smoother boundary conditions and larger area/perimeter ratios. For problems with discontinuous boundary data, the exponential convergence is also accomplished using the enriched method of fundamental solutions (EMFS), which is constructed by the fundamental solutions and the local singular solutions. The computation is scalable in the sense that the required time increases only algebraically.

Second-order elliptic PDEs with discontinuous boundary data

We shall consider the weak formulation of a linear elliptic model problem with discontinuous Dirichlet boundary conditions. Since such problems are typically not well-defined in the standard H^1-H^1 setting, we will introduce a suitable saddle point formulation in terms of weighted Sobolev spaces. Furthermore, we will discuss the numerical solution of such problems. Specifically, we employ an hp-discontinuous Galerkin method and derive an L^2-norm a posteriori error estimate. Numerical experiments demonstrate the effectiveness of the proposed error indicator in both the h- and hp-version setting. Indeed, in the latter case exponential convergence of the error is attained as the mesh is adaptively refined.

A meshless numerical method for solving slow mixed convections in containers with discontinuous boundary data

AbstractThis paper describes the formulations of the method of fundamental solutions (MFS), which is a famous meshless numerical method representing a sought solution by a series of fundamental solutions to solve slow mixed convections in containers with discontinuous boundary data. In the derivations, the fundamental solutions were obtained by using the Hörmander operator decomposition technique. All the velocities, temperatures, pressures, stresses and thermal fluxes corresponding to the fundamental solutions were addressed explicitly in tensor forms. Although the MFS is highly accurate for smooth boundary data, its convergence becomes poor when it is applied to problems with discontinuous boundary data. To compensate for this drawback, we enriched the MFS by adding the local discontinuous solutions to the series of fundamental solutions. This enriched MFS was applied to solve the benchmark problems of a lid‐driven cavity and natural convection in rectangular containers. In addition, the numerical solutions were compared with the analytical solutions. Then, the meshless numerical method was further utilized to solve mixed convections in a triangular cavity and a cavity with a cosine‐shaped bottom. These numerical results demonstrated the applicability of the enriched MFS to two‐dimensional mixed convections in containers with discontinuous boundary data. Copyright © 2010 John Wiley &amp; Sons, Ltd.

On existence of a classical solution and nonexistence of a weak solution to the Dirichlet problem for the Laplacian with discontinuous boundary data

The Dirichlet problem for the Laplacian in a planar multiply connected interior domain bounded by smooth closed curves is considered in the case when the boundary data is piecewise continuous; i.e., it may have jumps in certain points of the boundary. It is assumed that the solution to the problem may not be continuous at the same points. The well-posed formulation of the problem is given, theorems on existence and uniqueness of a classical solution are proved, and the integral representation for a classical solution is obtained. The problem is reduced to a uniquely solvable Fredholm integral equation of the second kind and of index zero. It is shown that a weak solution to the problem does not exist typically, though the classical solution exists.

A singular perturbation problem with discontinuous data in a cuboid

We analyse the asymptotic behaviour of the solution of a 3D singularly perturbed convection-diffusion problem with discontinuous Dirichlet boundary data defined in a cuboid. We write the solution in terms of a double series and we obtain an asymptotic approximation of the solution when the singular parameter e → 0. This approximation is given in terms of a finite combination of products of error functions and characterizes the effect of the discontinuities on the small e-behaviour of the solution in the singular layers.

The Dirichlet problem for the Laplacian with discontinuous boundary data in a 2D multiply connected exterior domain

The Dirichlet problem for Laplacian in a planar multiply connected exterior domain bounded by smooth closed curves is considered in case, when the boundary data is piecewise continuous, i.e. it may have jumps in certain points of the boundary. It is assumed that the solution to the problem may be not continuous at the same points. The well-posed formulation of the problem is given, theorems on existence and uniqueness of a classical solution are proved, the integral representation for a classical solution is obtained. The problem is reduced to a uniquely solvable Fredholm integral equation of the second kind and of index zero. It is shown that a weak solution to the problem does not exist typically, though the classical solution exists.

An error estimate for two-dimensional Stokes driven cavity flow

Discontinuous velocity boundary data for the lid driven cavity flow has long been causing difficulties in both theoretical analysis and numerical simulations. In finite element methods, the variational form for the driven cavity flow is not valid since the velocity is not in H 1 . Hence standard error estimates do not work. By using only W 1,r (1 < r < 2) regularity and constructing a continuous approximation to the boundary data, here we present error estimates for both the velocity-pressure formulation and the pseudostress-velocity formulation of the two-dimensional Stokes driven cavity flow.

Shock bifurcation and emergence of diffusive solitons in a nonlinear wave equation with relaxation

A hyperbolic generalization of Burgers' equation, which includes relaxation, is examined using analytical and numerical tools. By means of singular surface theory, the evolution of initial discontinuities (i.e. shocks) is fully classified. In addition, the parameter space is explored and the bifurcation experienced by the shock amplitude is identified. Then, by means of numerical simulations based on a Godunov-type scheme, we confirm the theoretical findings and explore the solution structure of a signaling-type initial-boundary-value problem with discontinuous boundary data. In particular, we show that diffusive solitons (or Taylor shocks) can emerge in the solution, behind the wavefront. We also show that, for certain parameter values, a shock wave becomes an acceleration wave in infinite time, an unexpected result that is the exact opposite of the well-known phenomenon of finite-time acceleration wave blow-up. Finally, the ‘red light turning green’ problem is re-examined.

ON ROBUST ON-LINE PARAMETER IDENTIFICATION TECHNIQUES IN ENVIRONMENTAL MODELING

In our paper we would like to discuss about one method of auxiliary controlled models and the application of this method to solving some problems of identification for differential equations. Problems of determining some parameters through the information on equation's solutions are often called reconstruction (identification) problems. Therewith it is assumed that the input information (results of measurements of current phase states of a dynamical system) forthcomes in the process. As to unknown parameters, they should be reconstructed in the process too. One of the methods of solving similar problems was suggested in (Kryazhimskii and Osipov, 1983; Kryazhimskii and Osipov, 1984; Osipov and Kryazhimskii, 1995; Maksimov, 1995; Maksimov, 2002a; Maksimov, 2002b). This method based on the ideas of the theory of ill-posed problems actually reduces an identification problem to a control problem for an auxiliary dynamical system—the model (Krasovskii and Subbotin, 1988). Regularization of the problem under consideration is locally realized during the process of choosing a positional control in the system-model. The method mentioned above was applied to a number of problems described by some classes of ordinary differential equations as well as by equations with distributed parameters. Different system's characteristics varying in time were under reconstruction, for example, unknown discontinuous inputs, initial and boundary data, distributed disturbances, coefficients of an elliptic operator and so on. In the present paper, we illustrate this method with a model example.

Spectral Element Discretization of the Circular Driven Cavity. Part IV: The Navier-Stokes Equations

This paper deals with the spectral element discretization of the Navier-Stokes equations in a disk with discontinuous boundary data, which is known as the driven cavity problem. The numerical treatment does not involve any regularization of these data. Relying on a variational formulation in the primitive variables of velocity and pressure, we describe a discretization of these equations and derive error estimates in appropriate weighted Sobolev spaces. We propose an algorithm to solve the nonlinear discrete system and present numerical experiments to verify its efficiency.

Spectral Element Discretization of the Circular Driven Cavity. Part III: The Stokes Equations in Primitive Variables

This paper deals with the spectral element discretization of the Stokes equations in a disk with discontinuous boundary data. The numerical treatment does not involve any regularization of these data. Based on a variational formulation in the primitive variables of velocity and pressure, we describe a discretization of the Stokes problem and derive error estimates in Sobolev spaces with appropriate weights. We propose an algorithm to solve the discrete system and present numerical experiments which confirm the efficiency of the method.

Spectral Element Discretization of the Circular Driven Cavity, Part II: The Bilaplacian Equation

This paper deals with the spectral element discretization of the bilaplacian equation in a disk with discontinuous boundary data. Relying on an appropriate variational formulation, we propose a discrete problem and prove its convergence. The use of weighted Sobolev spaces to treat the discontinuity of the boundary conditions also allows for improving the order of convergence. The results of the numerical experiments we present are in agreement with the theoretical ones.

Adaptive finite element computational fluid dynamics using an anisotropic error estimator

The aim of this paper is to present the results on finite element adaptive strategies for computational fluid dynamics (CFD) problems with singularities arising from shock phenomena and/or discontinuous boundary data. The adaptive analysis is based on an optimal-mesh-adaptive strategy which is employed to refine the mesh, stretch and orient the elements in such a way that, along the adaptation process, the mesh becomes aligned with the discontinuities. This mesh adaptation process yields improved results in locating regions of rapid or abrupt variations of the variables, whose location is not known a priori. On the other hand, the proposed mesh adaptation process is generated by minimizing, for a given number of elements in the mesh, a new anisotropic error estimator based on local directional interpolation error and recovering of the second derivatives of the finite element solution. Several adaptive mesh-refinement solutions for interpolation problems are presented in order to show that the proposed optimal adaptive strategy using this anisotropic error estimator recovers optimal and/or superconvergent rates. Finally, applications of this approach to CFD problems are also presented in order to show the computational performance of the proposed optimal adaptive procedure.