Inhomogeneous Dirichlet Boundary Conditions Research Articles

Finite Element Discretization of the Steady, Generalized Navier–Stokes Equations with Inhomogeneous Dirichlet Boundary Conditions

Finite Element Discretization of the Steady, Generalized Navier–Stokes Equations with Inhomogeneous Dirichlet Boundary Conditions

Solvability analysis for the Boussinesq model of heat transfer under the nonlinear Robin boundary condition for the temperature.

We consider the new boundary value problem for the generalized Boussinesq model of heat transfer under the inhomogeneous Dirichlet boundary condition for the velocity and under mixed boundary conditions for the temperature. It is assumed that the viscosity, thermal conductivity and buoyancy force in the model equations, as well as the heat exchange boundary coefficient, depend on the temperature. The mathematical apparatus for studying the inhomogeneous boundary value problem under study based on the variational method is being developed. Using this apparatus, we prove the main theorem on the global existence of a weak solution of the mentioned boundary value problem and establish sufficient conditions for the problem data ensuring the local uniqueness of the weak solution that has the additional property of smoothness with respect to temperature. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

Inhomogeneous Boundary Value Problems for the Generalized Boussinesq Model of Mass Transfer

We consider boundary value problems for a nonlinear mass transfer model, which generalizes the classical Boussinesq approximation, under inhomogeneous Dirichlet boundary conditions for the velocity and the substance’s concentration. It is assumed that the leading coefficients of viscosity and diffusion and the buoyancy force in the model equations depend on concentration. We develop a mathematical apparatus for studying the inhomogeneous boundary value problems under consideration. It is based on using a weak solution of the boundary value problem and on the construction of liftings of the inhomogeneous boundary data. They remove the inhomogeneity of the data and reduce initial problems to equivalent homogeneous boundary value problems. Based on this apparatus we will prove the theorem of the global existence of a weak solution to the boundary value problem under study and establish important properties of the solution. In particular, we will prove the validity of the maximum principle for the substance’s concentration. We will also establish sufficient conditions for the problem data, ensuring the local uniqueness of weak solutions.

Minimal residual methods in negative or fractional Sobolev norms

For numerical approximation the reformulation of a PDE as a residual minimisation problem has the advantages that the resulting linear system is symmetric positive definite, and that the norm of the residual provides an a posteriori error estimator. Furthermore, it allows for the treatment of general inhomogeneous boundary conditions. In many minimal residual formulations, however, one or more terms of the residual are measured in negative or fractional Sobolev norms. In this work, we provide a general approach to replace those norms by efficiently evaluable expressions without sacrificing quasi-optimality of the resulting numerical solution. We exemplify our approach by verifying the necessary inf-sup conditions for four formulations of a model second order elliptic equation with inhomogeneous Dirichlet and/or Neumann boundary conditions. We report on numerical experiments for the Poisson problem with mixed inhomogeneous Dirichlet and Neumann boundary conditions in an ultra-weak first order system formulation.

Dynamics and robustness for the 2D Navier–Stokes equations with multi-delays in Lipschitz-like domains

This paper is concerned with the dynamics of the two-dimensional Navier–Stokes equations with multi-delays in a Lipschitz-like domain, subject to inhomogeneous Dirichlet boundary conditions. The regularity of global solutions and of pullback attractors, based on tempered universes, is established, extending the results of Yang, Wang, Yan and Miranville (Discrete Contin. Dyn. Syst. 41 (2021) 3343–3366). Furthermore, the robustness of pullback attractors when the delays, considered as small perturbations, disappear is also derived. The key technique in the proofs is the application of a retarded Gronwall inequality and a variable index for the tempered pullback dynamics, allowing to obtain uniform estimates and the compactness of the process.

A (3+1)-dimensional equation of plasma drift waves perturbed by a singular potential in an infinite parallelepiped

We investigate the existence and nonexistence of weak solutions to a (3+1)-dimensional equation of plasma drift waves perturbed by a singular potential. The considered equation is posed in an infinite parallelepiped, under an inhomogeneous Dirichlet boundary condition. We show that the dividing line with respect to existence or nonexistence is given by a critical exponent.

Existence of a steady flow through a rotating radial turbine with an arbitrarily large inflow and an artificial boundary condition on the outflow

AbstractWe prove the existence of a steady strong solution to the Navier–Stokes‐type problem in a 2D multiply connected domain, modeling a flow of an incompressible viscous fluid through a rotating radial turbine. We consider the inhomogeneous Dirichlet boundary condition on the inflow Γin and an artificial boundary condition of the “do nothing” type on the outflow Γout. The conditions admit an arbitrarily large flux through the turbine. This is, however, compensated by the requirement that the distance between Γout and the profile family is “sufficiently small”. (Theorem 1.1.) The solution is steady in the rotating frame, that is, in the frame, attached to the rotating turbine. As an auxiliary result, we prove that a function, that takes just two different values in a 1D interval on two complementary measurable sets M and of positive 1D measure, is not in . (Lemma 3.4.)

Constraint Energy Minimizing Generalized Multiscale Finite Element Method for Inhomogeneous Boundary Value Problems with High Contrast Coefficients

In this article we develop the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) for elliptic partial differential equations with inhomogeneous Dirichlet, Neumann, and Robin boundary conditions, and the high contrast property emerges from the coefficients of elliptic operators and Robin boundary conditions. By careful construction of multiscale bases of the CEM-GMsFEM, we introduce two operators and which are used to handle inhomogeneous Dirichlet and Neumann boundary values and are also proved to converge independently of contrast ratios as enlarging oversampling regions. We providean a priori error estimate and show that the number of oversampling layers is the key factor in controlling numerical errors. A series of experiments are conducted, and those results reflect the reliability of our methods even with high contrast ratios.

Iterative Solution of a Retrospective Inverse Heat Conduction Problem with Inhomogeneous Dirichlet Boundary Conditions

Iterative Solution of a Retrospective Inverse Heat Conduction Problem with Inhomogeneous Dirichlet Boundary Conditions

Analytical solutions of the nitrogen uptake model with Michaelis-Menten flux

In this paper, we describe the application of the generalized finite Hankel transform method to the convection diffusion problems of nitrogen in soil subject to Robin boundary conditions and the right-hand side of the left Robin boundary condition is the classic nonlinear Michaelis-Menten flux. If the Michaelis-Menten flux is taken as a function of time, the analytical solution of series form can be derived by the generalized finite Hankel transform. In the calculation, the Michaelis-Menten flux is evaluated by the numerical concentration at the root surface and the number of terms of the partial sum is determined after adequately approximating the numerical solution. Therefore, we improve and develop the analytical method built by Garg et al., and exemplify and verify it by the nitrogen uptake model for roots. The investigated model has a representatively and completely mathematical structure among most nutrient uptake models, and then the analytical method and the solutions developed here can also be used to nutrient uptake problems in soil subject to homogeneous or inhomogeneous Dirichlet or Neumann boundary conditions.

A posteriori error estimators for discontinuous Galerkin method for diffusion problems, based on the hypercircle method

In this work, we apply the hypercircle method to Discontinuous Galerkin (DG) approximations of second order diffusion problems featuring inhomogeneous Dirichlet and Neumann boundary conditions. We focus on the interior penalty discontinuous Galerkin (IPDG) approximations of diffusion problems in primal variational formulation and produce a Prager–Synge theorem for such DG methods. Using the hypercircle method, we derive an a posteriori error estimator in terms of an equilibrated flux. The estimator is proven to be reliable and efficient. Numerical results are presented which illustrate the estimator’s performance.

Finite Difference Method for Inhomogeneous Fractional Dirichlet Problem

We make the split of the integral fractional Laplacian as $(-\Delta)^s u=(-\Delta)(-\Delta)^{s-1}u$, where $s\in(0,\frac{1}{2})\cup(\frac{1}{2},1)$. Based on this splitting, we respectively discretize the one- and two-dimensional integral fractional Laplacian with the inhomogeneous Dirichlet boundary condition and give the corresponding truncation errors with the help of the interpolation estimate. Moreover, the suitable corrections are proposed to guarantee the convergence in solving the inhomogeneous fractional Dirichlet problem and an $\mathcal{O}(h^{1+\alpha-2s})$ convergence rate is obtained when the solution $u\in C^{1,\alpha}(\bar{\Omega}^{\delta}_{n})$, where $n$ is the dimension of the space, $\alpha\in(\max(0,2s-1),1]$, $\delta$ is a fixed positive constant, and $h$ denotes mesh size. Finally, the performed numerical experiments confirm the theoretical results.

Asymptotic stability of solutions to the Navier–Stokes–Fourier system driven by inhomogeneous Dirichlet boundary conditions

We consider global in time solutions of the Navier–Stokes–Fourier system describing the motion of a general compressible, viscous and heat conducting fluid far from equilibirum. Using a new concept of weak solution suitable to accommodate the inhomogeneous Dirichlet time dependent data we find sufficient conditions for the global in time weak solutions to be ultimately bounded.

Analytical solution of a non-homogeneous boundary-value problem for the transport equation in an earth to air energy exchanger by initial base analysis method

In this paper, a one-dimensional time-dependent transport equation with inhomogeneous Dirichlet boundary conditions, often encountered in the study of an Earth to Air Energy Exchanger, is solved by a practical analytical method. The initial condition function is not randomly set to verify the transport equation at the initial time, but determined as a first approximation to the solution of the transport equation (at t=0). As in the homotopy analysis method, the solution of the transport equation is expressed by the set of a basis function constructed with the obtained initial condition function. The components of the solution in this base, are time’s functions deduced from the boundary conditions of the problem. Suitable endings are noticeable between the proposed model and experimental measurements.

Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks

In this paper, we introduce a new approach based on distance fields to exactly impose boundary conditions in physics-informed deep neural networks. The challenges in satisfying Dirichlet boundary conditions in meshfree and particle methods are well-known. This issue is also pertinent in the development of physics informed neural networks (PINN) for the solution of partial differential equations. We introduce geometry-aware trial functions in artificial neural networks to improve the training in deep learning for partial differential equations. To this end, we use concepts from constructive solid geometry (R-functions) and generalized barycentric coordinates (mean value potential fields) to construct ϕ(x), an approximate distance function to the boundary of a domain in Rd. To exactly impose homogeneous Dirichlet boundary conditions, the trial function is taken as ϕ(x) multiplied by the PINN approximation, and its generalization via transfinite interpolation is used to a priori satisfy inhomogeneous Dirichlet (essential), Neumann (natural), and Robin boundary conditions on complex geometries. In doing so, we eliminate modeling error associated with the satisfaction of boundary conditions in a collocation method and ensure that kinematic admissibility is met pointwise in a Ritz method. With this new ansatz, the training for the neural network is simplified: sole contribution to the loss function is from the residual error at interior collocation points where the governing equation is required to be satisfied. Numerical solutions are computed using strong form collocation and Ritz minimization. To convey the main ideas and to assess the accuracy of the approach, we present numerical solutions for linear and nonlinear boundary-value problems over convex and nonconvex polygonal domains as well as over domains with curved boundaries. Benchmark problems in one dimension for linear elasticity, advection-diffusion, and beam bending; and in two dimensions for the steady-state heat equation, Laplace equation, biharmonic equation (Kirchhoff plate bending), and the nonlinear Eikonal equation are considered. The construction of approximate distance functions using R-functions extends to higher dimensions, and we showcase its use by solving a Poisson problem with homogeneous Dirichlet boundary conditions over the four-dimensional hypercube. The proposed approach consistently outperforms a standard PINN-based collocation method, which underscores the importance of exactly (a priori) satisfying the boundary condition when constructing a loss function in PINN. This study provides a pathway for meshfree analysis to be conducted on the exact geometry without domain discretization.

Galilean-Invariant Characteristic-Based Volume Penalization Method for Supersonic Flows with Moving Boundaries

This work extends the characteristic-based volume penalization method, originally developed and demonstrated for compressible subsonic viscous flows in (J. Comput. Phys. 262, 2014), to a hyperbolic system of partial differential equations involving complex domains with moving boundaries. The proposed methodology is shown to be Galilean-invariant and can be used to impose either homogeneous or inhomogeneous Dirichlet, Neumann, and Robin type boundary conditions on immersed boundaries. Both integrated and non-integrated variables can be treated in a systematic manner that parallels the prescription of exact boundary conditions with the approximation error rigorously controlled through an a priori penalization parameter. The proposed approach is well suited for use with adaptive mesh refinement, which allows adequate resolution of the geometry without over-resolving flow structures and minimizing the number of grid points inside the solid obstacle. The extended Galilean-invariant characteristic-based volume penalization method, while being generally applicable to both compressible Navier–Stokes and Euler equations across all speed regimes, is demonstrated for a number of supersonic benchmark flows around both stationary and moving obstacles of arbitrary shape.

Error Analysis of Nitsche’s and Discontinuous Galerkin Methods of a Reduced Landau–de Gennes Problem

Abstract We study a system of semi-linear elliptic partial differential equations with a lower order cubic nonlinear term, and inhomogeneous Dirichlet boundary conditions, relevant for two-dimensional bistable liquid crystal devices, within a reduced Landau–de Gennes framework. The main results are (i) a priori error estimates for the energy norm, within the Nitsche’s and discontinuous Galerkin frameworks under milder regularity assumptions on the exact solution and (ii) a reliable and efficient a posteriori analysis for a sufficiently large penalization parameter and a sufficiently fine triangulation in both cases. Numerical examples that validate the theoretical results, are presented separately.

Duality between range and no-response tests and its application for inverse problems

In this paper we will show the duality between the range test (RT) and no-response test (NRT) for the inverse boundary value problem for the Laplace equation in \begin{document}$ \Omega\setminus\overline D $\end{document} with an unknown obstacle \begin{document}$ D\Subset\Omega $\end{document} whose boundary \begin{document}$ \partial D $\end{document} is visible from the boundary \begin{document}$ \partial\Omega $\end{document} of \begin{document}$ \Omega $\end{document} and a measurement is given as a set of Cauchy data on \begin{document}$ \partial\Omega $\end{document} . Here the Cauchy data is given by a unique solution \begin{document}$ u $\end{document} of the boundary value problem for the Laplace equation in \begin{document}$ \Omega\setminus\overline D $\end{document} with homogeneous and inhomogeneous Dirichlet boundary condition on \begin{document}$ \partial D $\end{document} and \begin{document}$ \partial\Omega $\end{document} , respectively. These testing methods are domain sampling methods to estimate the location of the obstacle using test domains and the associated indicator functions. Also both of these testing methods can test the analytic extension of \begin{document}$ u $\end{document} to the exterior of a test domain. Since these methods are defined via some operators which are dual to each other, we could expect that there is a duality between the two methods. We will give this duality in terms of the equivalence of the pre-indicator functions associated to their indicator functions. As an application of the duality, the reconstruction of \begin{document}$ D $\end{document} using the RT gives the reconstruction of \begin{document}$ D $\end{document} using the NRT and vice versa. We will also give each of these reconstructions without using the duality if the Dirichlet data of the Cauchy data on \begin{document}$ \partial\Omega $\end{document} is not identically zero and the solution to the associated forward problem does not have any analytic extension across \begin{document}$ \partial D $\end{document} . Moreover, we will show that these methods can still give the reconstruction of \begin{document}$ D $\end{document} if we a priori knows that \begin{document}$ D $\end{document} is a convex polygon and it satisfies one of the following two properties: all of its corner angles are irrational and its diameter is less than its distance to \begin{document}$ \partial\Omega $\end{document} .

Imposing nonlocal boundary conditions in Galerkin-type methods based on non-interpolatory functions

The imposition of inhomogeneous Dirichlet (essential) boundary conditions is a fundamental challenge in the application of Galerkin-type methods based on non-interpolatory functions, i.e., functions which do not possess the Kronecker delta property. Such functions typically are used in various meshfree methods, as well as methods based on the isogeometric paradigm. The present paper analyses a model problem consisting of the Poisson equation subject to non-standard boundary conditions. Namely, instead of classical boundary conditions, the model problem involves Dirichlet- and Neumann-type nonlocal boundary conditions. Variational formulations with strongly and weakly imposed inhomogeneous Dirichlet-type nonlocal conditions are derived and compared within an extensive numerical study in the isogeometric framework based on non-uniform rational B-splines (NURBS). The attention in the numerical study is paid mainly to the influence of the nonlocal boundary conditions on the properties of the considered discretisation methods.

Error analysis for the finite element approximation of the Darcy–Brinkman–Forchheimer model for porous media with mixed boundary conditions

This paper deals with the finite element approximation of the Darcy–Brinkman- Forchheimer equation, involving a porous media with spatially-varying porosity, with mixed boundary condition such as inhomogeneous Dirichlet and traction boundary conditions. We first prove that the considered problem has a unique solution if the source terms are small enough. The convergence of a Taylor–Hood finite element approximation using a finite element interpolation of the porosity is then proved under similar smallness assumptions. Some optimal error estimates are obtained if the solution to the Darcy–Brinkman–Forchheimer model is smooth enough. We end this paper by providing a fixed-point method to solve the discrete non-linear problems and with some numerical experiments to make more precise the smallness assumptions on the source terms and to illustrate the theoretical convergence results.