The author proves that there exist three solutions u(0), u(1) and u(2) in the following problem [GRAPHICS] where some conditions are imposed on Q and f. Here, 0 < u(0) < u(1), u(2) changes sign.
In this paper we study the existence of nontrivial solutions of a class of asymptotically linear elliptic resonant problems at higher eigenvalues with the nonlinear term which may be un- bounded by making use of the M...In this paper we study the existence of nontrivial solutions of a class of asymptotically linear elliptic resonant problems at higher eigenvalues with the nonlinear term which may be un- bounded by making use of the Morse theory for a C^2-function at both isolated critical point and infinity.展开更多
We give an existence result of entropy and renormalized solutions for strongly nonlinear elliptic equations in the framework of Sobolev spaces with variable exponents of the type： -div （a（x, u,▽u）＋φ（u））＋g（...We give an existence result of entropy and renormalized solutions for strongly nonlinear elliptic equations in the framework of Sobolev spaces with variable exponents of the type： -div （a（x, u,▽u）＋φ（u））＋g（x, u,▽u）=μ, where the right-hand side belongs to L^1（Ω）＋W^-1,p＇（x）（Ω）, -div（a（x, u,▽u）） is a Leray-Lions operator defined from W^-1,p＇（x）（Ω） into its dual and φ∈C^0（R,R^N）. The function g（x, u,▽u） is a non linear lower order term with natural growth with respect to ｜▽u｜ satisfying the sign condition, that is, g（x, u,▽u）u ≥ 0.展开更多
We design and analyze an iterative two-grid algorithm for the finite element discretizations of strongly nonlinear elliptic boundary value problems in this paper.We propose an iterative two-grid algorithm,in which a n...We design and analyze an iterative two-grid algorithm for the finite element discretizations of strongly nonlinear elliptic boundary value problems in this paper.We propose an iterative two-grid algorithm,in which a nonlinear problem is first solved on the coarse space,and then a symmetric positive definite problem is solved on the fine space.The main contribution in this paper is to establish a first convergence analysis,which requires dealing with four coupled error estimates,for the iterative two-grid methods.We also present some numerical experiments to confirm the efficiency of the proposed algorithm.展开更多
Abstract In this article, we investigate the equations of magnetostaties for a configuration where a ferromagnetic material occupies a bounded domain and is surrounded by vacuum. Furthermore, the ferromagnetic law tak...Abstract In this article, we investigate the equations of magnetostaties for a configuration where a ferromagnetic material occupies a bounded domain and is surrounded by vacuum. Furthermore, the ferromagnetic law takes the form B=μ0μr（｜H｜）Hi i.e., the magnetizing field H and the magnetic induction B are collinear, but the relative permeability μr is allowed to depend on the modulus of H. We prove the well-posedness of the magnetostatic problem under suitable convexity assumptions, and the convergence of several iterative methods, both for the original problem set in the Beppo-Levi space W1（R3）, and for a finite-dimensional approximation. The theoretical results are illustrated by numerical examples, which capture the known physical phenomena.展开更多
We study galvanic currents on a heterogeneous surface. In electrochemistry, the oxidation-reduction reaction producing the current is commonly modeled by a nonlinear elliptic boundary value problem. The boundary condi...We study galvanic currents on a heterogeneous surface. In electrochemistry, the oxidation-reduction reaction producing the current is commonly modeled by a nonlinear elliptic boundary value problem. The boundary condition is of exponential type with periodically varying parameters. We construct an approximation by first homogenizing the problem, and then linearizing about the homogenized solution. This approximation is far more accurate than both previous approximations or direct linearization. We establish convergence estimates for both the two and three-dimensional case and provide two-dimensional numerical experiments.展开更多
We examine a steady-state heat radiation problem and its finite element approximation in Rd, d = 2, 3. A nonlinear Stefan-Boltzmann boundary condition is considered. Another nonlinearity is due to the fact that the te...We examine a steady-state heat radiation problem and its finite element approximation in Rd, d = 2, 3. A nonlinear Stefan-Boltzmann boundary condition is considered. Another nonlinearity is due to the fact that the temperature is always greater or equal than 0[K]. We prove two convergence theorems for piecewise linear finite element solutions.展开更多
文摘The author proves that there exist three solutions u(0), u(1) and u(2) in the following problem [GRAPHICS] where some conditions are imposed on Q and f. Here, 0 < u(0) < u(1), u(2) changes sign.
文摘In this paper we study the existence of nontrivial solutions of a class of asymptotically linear elliptic resonant problems at higher eigenvalues with the nonlinear term which may be un- bounded by making use of the Morse theory for a C^2-function at both isolated critical point and infinity.
文摘We give an existence result of entropy and renormalized solutions for strongly nonlinear elliptic equations in the framework of Sobolev spaces with variable exponents of the type： -div （a（x, u,▽u）＋φ（u））＋g（x, u,▽u）=μ, where the right-hand side belongs to L^1（Ω）＋W^-1,p＇（x）（Ω）, -div（a（x, u,▽u）） is a Leray-Lions operator defined from W^-1,p＇（x）（Ω） into its dual and φ∈C^0（R,R^N）. The function g（x, u,▽u） is a non linear lower order term with natural growth with respect to ｜▽u｜ satisfying the sign condition, that is, g（x, u,▽u）u ≥ 0.
基金funded by the Science and Technology Development Fund,Macao SAR(Grant Nos.0070/2019/A2,0031/2022/A1)supported by the National Natural Science Foundation of China(Grant No.11901212)supported by the National Natural Science Foundation of China(Grant No.12071160).
文摘We design and analyze an iterative two-grid algorithm for the finite element discretizations of strongly nonlinear elliptic boundary value problems in this paper.We propose an iterative two-grid algorithm,in which a nonlinear problem is first solved on the coarse space,and then a symmetric positive definite problem is solved on the fine space.The main contribution in this paper is to establish a first convergence analysis,which requires dealing with four coupled error estimates,for the iterative two-grid methods.We also present some numerical experiments to confirm the efficiency of the proposed algorithm.
文摘Abstract In this article, we investigate the equations of magnetostaties for a configuration where a ferromagnetic material occupies a bounded domain and is surrounded by vacuum. Furthermore, the ferromagnetic law takes the form B=μ0μr（｜H｜）Hi i.e., the magnetizing field H and the magnetic induction B are collinear, but the relative permeability μr is allowed to depend on the modulus of H. We prove the well-posedness of the magnetostatic problem under suitable convexity assumptions, and the convergence of several iterative methods, both for the original problem set in the Beppo-Levi space W1（R3）, and for a finite-dimensional approximation. The theoretical results are illustrated by numerical examples, which capture the known physical phenomena.
基金Acknowledgments. This research is partially supported by the National Science Foundation Grants DMS#0619080 and DMS#0605021.
文摘We study galvanic currents on a heterogeneous surface. In electrochemistry, the oxidation-reduction reaction producing the current is commonly modeled by a nonlinear elliptic boundary value problem. The boundary condition is of exponential type with periodically varying parameters. We construct an approximation by first homogenizing the problem, and then linearizing about the homogenized solution. This approximation is far more accurate than both previous approximations or direct linearization. We establish convergence estimates for both the two and three-dimensional case and provide two-dimensional numerical experiments.
文摘We examine a steady-state heat radiation problem and its finite element approximation in Rd, d = 2, 3. A nonlinear Stefan-Boltzmann boundary condition is considered. Another nonlinearity is due to the fact that the temperature is always greater or equal than 0[K]. We prove two convergence theorems for piecewise linear finite element solutions.
