Coercive Problems Research Articles

Nonlinear Dirichlet problems with unilateral growth on the reaction

Abstract We consider a nonlinear Dirichlet problem driven by the p-Laplace differential operator with a reaction which has a subcritical growth restriction only from above. We prove two multiplicity theorems producing three nontrivial solutions, two of constant sign and the third nodal. The two multiplicity theorems differ on the geometry near the origin. In the semilinear case (that is, p = 2 {p=2} ), using Morse theory (critical groups), we produce a second nodal solution for a total of four nontrivial solutions. As an illustration, we show that our results incorporate and significantly extend the multiplicity results existing for a class of parametric, coercive Dirichlet problems.

Continuous Dependence on Translations of the Independent Variable for Solutions of Boundary-Value Problems for Differential-Difference Equations

We consider boundary-value problems for differential-difference operators with perturbations in translations of the independent variable. We prove that the family of differential-difference operators is positive definite uniformly with respect to translations of the independent variable. Solutions of such problems depend continuously on these translations. We consider the coercivity problem for differential-difference operators with incommensurable translations of the independent variable and study the approximation of such operators by rational operators.

The aggregated unfitted finite element method for elliptic problems

Unfitted finite element techniques are valuable tools in different applications where the generation of body-fitted meshes is difficult. However, these techniques are prone to severe ill conditioning problems that obstruct the efficient use of iterative Krylov methods and, in consequence, hindersthe practical usage of unfitted methods for realistic large scale applications. In this work, we present a technique that addresses such conditioning problems by constructing enhanced finite element spaces based on a cell aggregation technique. The presented method, called aggregated unfitted finite element method, is easy to implement, and can be used, in contrast to previous works, in Galerkin approximations of coercive problems with conforming Lagrangian finite element spaces. The mathematical analysis of the method states that the condition number of the resulting linear system matrix scales as in standard finite elements for body-fitted meshes, without being affected by small cut cells, and that the method leads to the optimal finite element convergence order. These theoretical results are confirmed with 2D and 3D numerical experiments.

Mixed Aggregated Finite Element Methods for the Unfitted Discretization of the Stokes Problem

In this work, we consider unfitted finite element methods for the numerical approximation of the Stokes problem. It is well-known that this kind of methods lead to arbitrarily ill-conditioned systems. In order to solve this issue, we consider the recently proposed aggregated finite element method, originally motivated for coercive problems. However, the well-posedness of the Stokes problem is far more subtle and relies on a discrete inf-sup condition. We consider mixed finite element methods that satisfy the discrete version of the inf-sup condition for body-fitted meshes, and analyze how the discrete inf-sup is affected when considering the unfitted case. We propose different aggregated mixed finite element spaces combined with simple stabilization terms, which can include pressure jumps and/or cell residuals, to fix the potential deficiencies of the aggregated inf-sup. We carry out a complete numerical analysis, which includes stability, optimal a priori error estimates, and condition number bounds that are not affected by the small cut cell problem. For the sake of conciseness, we have restricted the analysis to hexahedral meshes and discontinuous pressure spaces. A thorough numerical experimentation bears out the numerical analysis. The aggregated mixed finite element method is ultimately applied to two problems with non-trivial geometries.

Multiple solutions for nonlinear nonhomogeneous resonant coercive problems

We consider a nonlinear, nonhomogeneous Dirichlet problem driven by the sum of a \begin{document}$p$\end{document} -Laplacian ( \begin{document}$2 ) and a Laplacian. The reaction term is a Caratheodory function \begin{document}$f(z,x)$\end{document} which is resonant with respect to the principal eigenvalue of ( \begin{document}$-\Delta_p,\, W^{1,p}_0(\Omega)$\end{document} ). Using variational methods combined with truncation and comparison techniques and Morse theory (critical groups) we prove the existence of three nontrivial smooth solutions all with sign information and under three different conditions concerning the behavior of \begin{document}$f(z,\cdot)$\end{document} near zero. By strengthening the regularity of \begin{document}$f(z,\cdot)$\end{document} , we are able to generate a second nodal solution for a total of four nontrivial smooth solutions.

Alternating direction method of multiplier for a unilateral contact problem in electro-elastostatics

We study an alternating direction method of multiplier (ADMM) applied to a unilateral frictional contact problem between an electro-elastic material and an electrically non conductive foundation. The frictional contact is modeled by the Tresca friction law. The resulting coupled problem is non symmetric and non coercive. By eliminating the electric potential, we obtain a symmetric and coercive problem which can be reformulated as a convex minimization problem. We then apply an alternating direction method of multiplier for the numerical approximation. To avoid explicit matrices inverse (due to the elimination of the electric potential) we use a preconditioned conjugate gradient algorithm as an inner solver. Numerical experiments are proposed to illustrate the efficiency of the proposed approach.

Least-Squares Proper Generalized Decompositions for Weakly Coercive Elliptic Problems

Proper generalized decompositions (PGDs) are a family of methods for efficiently solving high-dimensional PDEs, which seek to find a low-rank approximation to the solution of the PDE a priori. Convergence of PGD algorithms can only be proven for problems which are continuous, symmetric, and strongly coercive. In the particular case of problems which are only weakly coercive we have the additional issue that weak coercivity estimates are not guaranteed to be inherited by the low-rank PGD approximation. This can cause stability issues when employing a Galerkin PGD approximation of weakly coercive problems. In this paper we propose the use of PGD algorithms based on least-squares formulations which always lead to symmetric and strongly coercive problems and hence provide stable and provably convergent algorithms. Taking the Stokes problem as a prototypical example of a weakly coercive problem, we develop and compare rigorous least-squares PGD algorithms based on continuous least-squares estimates for two dif...

無潤滑ディスク DLC の摩擦特性の温度依存性に関する研究

To further increase recording density of disk storage drives, the heat assisted magnetic recording (HAMR) can solve the coercive force problem of thermal fluctuation and is regarded as the key technology for achieving recording density of more than 1Tb/in2 (1)(2). In this technology, it is suggested that there are serious issues for the diamond like carbon (DLC) overcoat on the disk surface. Because that is heated to high temperature with laser beams(2). In this study, first, we measured temperature dependence of friction properties of non-lubricated DLC surface on magnetic disks. Second, we considered the influence of the accumulated smear on test pin surface during the friction test. As a result, it was estimated that the smear on pin surface accumulated by the laser irradiation was originated from the hydrocarbon contamination adsorbed on DLC surface.

Eliminating the pollution effect in Helmholtz problems by local subscale correction

We introduce a new Petrov-Galerkin multiscale method for the numerical approximation of the Helmholtz equation with large wave number $\kappa$ in bounded domains in $\mathbb{R}^d$. The discrete trial and test spaces are generated from standard mesh-based finite elements by local subscale corrections in the spirit of numerical homogenization. The precomputation of the corrections involves the solution of coercive cell problems on localized subdomains of size $\ell H$; $H$ being the mesh size and $\ell$ being the oversampling parameter. If the mesh size and the oversampling parameter are such that $H\kappa$ and $\log(\kappa)/\ell$ fall below some generic constants and if the cell problems are solved sufficiently accurate on some finer scale of discretization, then the method is stable and its error is proportional to $H$; pollution effects are eliminated in this regime.

Noncoercive Resonant (p, 2)-Equations

We consider a nonlinear Dirichlet problem driven by the sum of p-Laplacian and a Laplacian (a (p, 2)-equation) which is resonant at \(\pm \infty \) with respect to the principal eigenvalue \(\hat{\lambda }_1(p)\) of \((-\Delta _p,W^{1,p}_{0}(\Omega ))\) and resonant at zero with respect to any nonprincipal eigenvalue of \((-\Delta ,H^1_0(\Omega ))\). At \(\pm \infty \) the resonance occurs from the right of \(\hat{\lambda }_1(p)\) and so the energy functional of the problem is indefinite. Using critical groups, we show that the problem has at least one nontrivial smooth solution. The result complements the recent work of Papageorgiou and Rădulescu (Appl Math Optim 69:393–430, 2014), where resonant (p, 2)-equations were examined with the resonance occurring from the left of \(\hat{\lambda }_1(p)\) (coercive problem).

Functional A Posteriori Error Control for Conforming Mixed Approximations of Coercive Problems with Lower Order Terms

Abstract The results of this contribution are derived in the framework of functional type a posteriori error estimates. The error is measured in a combined norm which takes into account both the primal and dual variables denoted by x and y, respectively. Our first main result is an error equality for all equations of the class A * ⁢ A ⁢ x + x = f ${\mathrm{A}^{*}\mathrm{A}x+x=f}$ or in mixed formulation A * ⁢ y + x = f ${\mathrm{A}^{*}y+x=f}$ , A ⁢ x = y ${\mathrm{A}x=y}$ , where the exact solution ( x , y ) $(x,y)$ is in D ⁢ ( A ) × D ⁢ ( A * ) $D(\mathrm{A})\times D(\mathrm{A}^{*})$ . Here A ${\mathrm{A}}$ is a linear, densely defined and closed (usually a differential) operator and A * ${\mathrm{A}^{*}}$ its adjoint. In this paper we deal with very conforming mixed approximations, i.e., we assume that the approximation ( x ~ , y ~ ) ${(\tilde{x},\tilde{y})}$ belongs to D ⁢ ( A ) × D ⁢ ( A * ) ${D(\mathrm{A})\times D(\mathrm{A}^{*})}$ . In order to obtain the exact global error value of this approximation one only needs the problem data and the mixed approximation itself, i.e., we have the equality | x - x ~ | 2 + | A ⁢ ( x - x ~ ) | 2 + | y - y ~ | 2 + | A * ⁢ ( y - y ~ ) | 2 = ℳ ⁢ ( x ~ , y ~ ) , $\lvert x-\tilde{x}\rvert^{2}+\lvert\mathrm{A}(x-\tilde{x})\rvert^{2}+\lvert y-% \tilde{y}\rvert^{2}+\lvert\mathrm{A}^{*}(y-\tilde{y})\rvert^{2}=\mathcal{M}(% \tilde{x},\tilde{y}),$ where ℳ ⁢ ( x ~ , y ~ ) := | f - x ~ - A * ⁢ y ~ | 2 + | y ~ - A ⁢ x ~ | 2 ${\mathcal{M}(\tilde{x},\tilde{y}):=\lvert f-\tilde{x}-\mathrm{A}^{*}\tilde{y}% \rvert^{2}+\lvert\tilde{y}-\mathrm{A}\tilde{x}\rvert^{2}}$ contains only known data. Our second main result is an error estimate for all equations of the class A * ⁢ A ⁢ x + i ⁢ x = f ${\mathrm{A}^{*}\mathrm{A}x+ix=f}$ or in mixed formulation A * ⁢ y + i ⁢ x = f ${\mathrm{A}^{*}y+ix=f}$ , A ⁢ x = y ${\mathrm{A}x=y}$ , where i is the imaginary unit. For this problem we have the two-sided estimate 2 2 + 1 ⁢ ℳ i ⁢ ( x ~ , y ~ ) ≤ | x - x ~ | 2 + | A ⁢ ( x - x ~ ) | 2 + | y - y ~ | 2 + | A * ⁢ ( y - y ~ ) | 2 ≤ 2 2 - 1 ⁢ ℳ i ⁢ ( x ~ , y ~ ) , $\frac{\sqrt{2}}{\sqrt{2}+1}\mathcal{M}_{i}(\tilde{x},\tilde{y})\leq\lvert x-% \tilde{x}\rvert^{2}+\lvert\mathrm{A}(x-\tilde{x})\rvert^{2}+\lvert y-\tilde{y}% \rvert^{2}+\lvert\mathrm{A}^{*}(y-\tilde{y})\rvert^{2}\leq\frac{\sqrt{2}}{% \sqrt{2}-1}\mathcal{M}_{i}(\tilde{x},\tilde{y}),$ where ℳ i ⁢ ( x ~ , y ~ ) := | f - i ⁢ x ~ - A * ⁢ y ~ | 2 + | y ~ - A ⁢ x ~ | 2 ${\mathcal{M}_{i}(\tilde{x},\tilde{y}):=\lvert f-i\tilde{x}-\mathrm{A}^{*}% \tilde{y}\rvert^{2}+\lvert\tilde{y}-\mathrm{A}\tilde{x}\rvert^{2}}$ contains only known data. We will point out a motivation for the study of the latter problems by time discretizations or time-harmonic ansatz of linear partial differential equations and we will present an extensive list of applications including the reaction-diffusion problem and the eddy current problem.

Optimization-based additive decomposition of weakly coercive problems with applications

We present an abstract mathematical framework for an optimization-based additive decomposition of a large class of variational problems into a collection of concurrent subproblems. The framework replaces a given monolithic problem by an equivalent constrained optimization formulation in which the subproblems define the optimization constraints and the objective is to minimize the mismatch between their solutions. The significance of this reformulation stems from the fact that one can solve the resulting optimality system by an iterative process involving only solutions of the subproblems. Consequently, assuming that stable numerical methods and efficient solvers are available for every subproblem, our reformulation leads to robust and efficient numerical algorithms for a given monolithic problem by breaking it into subproblems that can be handled more easily. An application of the framework to the Oseen equations illustrates its potential.

Temperature-dependent magnetization reversal process and coercivity mechanism in Nd-Fe-B hot-deformed magnets

Low coercivity and its large temperature dependence of a Nd2Fe14B magnet with respect to its magnetic anisotropy field have been addressed as the coercivity problem. To elucidate the physical origin of this problem, we have investigated the temperature dependence of the magnetization reversal behavior in the Nd-Fe-B hot-deformed magnet. Based on the analysis of the energy barrier evaluated from magnetic viscosity measurements, the coercivity problem is discussed in terms of the following three aspects: magnetization reversal process, intrinsic coercivity without thermal demagnetization effect, and energy barrier height. The analyses lead us to conclude that domain wall pinning is dominant in the magnetization reversal in the Nd-Fe-B hot-deformed magnet. The temperature dependences of the intrinsic coercivity and the energy barrier height are explained by the grain boundary model with an intermediate layer. These analyses would be utilized to discuss the detailed structure and magnetic properties of the grain boundary, which gives a new insight to overcome the coercivity problem.

Coercive and noncoercive nonlinear Neumann problems with indefinite potential

Abstract We consider nonlinear Neumann problems driven by a nonhomogeneous differential operator and an indefinite potential. In this paper we are concerned with two distinct cases. We first consider the case where the reaction is (p-1)-sublinear near ±∞ and (p-1)-superlinear near zero. In this setting the energy functional of the problem is coercive. In the second case, the reaction is (p-1)-superlinear near ±∞ (without satisfying the Ambrosetti–Rabinowitz condition) and has a (p-1)-sublinear growth near zero. Now, the energy functional is indefinite. For both cases we prove “three solutions theorems” and in the coercive setting we provide sign information for all of them. Our approach combines variational methods, truncation and perturbation techniques, and Morse theory (critical groups).

Energy barrier analysis of Nd-Fe-B thin films

The magnetization reversal mechanism of a permanent magnet has long been a controversial issue, which is closely related to the so-called coercivity problem. It is well known that the energy barrier for magnetization reversal contains essential information on reversal process. In this study, we propose a method to analyze the energy barrier function for the magnetization reversal. Preferentially (001) oriented Nd-Fe-B films with and without a Nd overlayer are used as model magnets. By combining the magnetic viscosity and time dependent coercivity measurements, the barrier function has been successfully evaluated. As a result, although the Nd-Fe-B films with and without Nd overlayer exhibit different magnetic behaviors, the power indices for their energy barrier are almost the same, suggesting that the magnetization reversal proceeds in a similar mode.

A tensor approximation method based on ideal minimal residual formulations for the solution of high-dimensional problems

In this paper, we propose a method for the approximation of the solution of high-dimensional weakly coercive problems formulated in tensor spaces using low-rank approximation formats. The method can be seen as a perturbation of a minimal residual method with residual norm corresponding to the error in a specified solution norm. We introduce and analyze an iterative algorithm that is able to provide a controlled approximation of the optimal approximation of the solution in a given low-rank subset, without any a priori information on this solution. We also introduce a weak greedy algorithm which uses this perturbed minimal residual method for the computation of successive greedy corrections in small tensor subsets. We prove its convergence under some conditions on the parameters of the algorithm. The residual norm can be designed such that the resulting low-rank approximations are quasi-optimal with respect to particular norms of interest, thus yielding to goal-oriented order reduction strategies for the approximation of high-dimensional problems. The proposed numerical method is applied to the solution of a stochastic partial differential equation which is discretized using standard Galerkin methods in tensor product spaces.

On asymptotic exit-time control problems lacking coercivity

The research on a class of asymptotic exit-time problems with a vanishing Lagrangian, begun in [M. Motta and C. Sartori, Nonlinear Differ. Equ. Appl. Springer (2014).] for the compact control case, is extended here to the case of unbounded controls and data, including both coercive and non-coercive problems. We give sufficient conditions to have a well-posed notion of generalized control problem and obtain regularity, characterization and approximation results for the value function of the problem.

Unification of distance inequalities for linear variational problems

In this work a unifying approach is presented that leads to bounds for the distance in natural norms between solutions belonging to different spaces, of well-posed linear variational problems with the same input data. This is done in a general hilbertian framework, and in this sense, well-known inequalities such as Cea’s or Babuska’s for coercive and non-coercive problems are extended and/or refined, as mere by-products of this unified setting. More particularly such an approach gives rise to both an improvement and a generalization to the weakly coercive case, of second Strang’s inequality for abstract coercive problems. Additionally several aspects specific to linear variational problems are the subject of a thorough analysis beforehand, which also allows for clarifications and further refinements about the concept of weak coercivity.

On a second order coercive dirichlet problem with a non-differentiable action functional

Applying certain convexity arguments we investigate the existence of a classical solution for a Dirichlet problem for which the Euler action functional is not necessarily differentiable in the sense of Gâteaux.

Some coercive problems for the system of Poisson equations

In the paper, two boundary value problems for the system of Poisson equations in three-dimensional domains are studied.