In this paper,we will study the nonlocal and nonvariational elliptic problem{−(1+a||u||_(q)^(αq))Δu=|u|^(p−1)u+h(x,u,∇_(u))inΩ,u=0 on∂Ω,(0.1)(1)where a>0,α>0,1<q<2^(∗),p∈(0,2^(∗)−1)∖{1}andΩis a boun...In this paper,we will study the nonlocal and nonvariational elliptic problem{−(1+a||u||_(q)^(αq))Δu=|u|^(p−1)u+h(x,u,∇_(u))inΩ,u=0 on∂Ω,(0.1)(1)where a>0,α>0,1<q<2^(∗),p∈(0,2^(∗)−1)∖{1}andΩis a bounded smooth domain in R^(N)(N≥2).Under suitable assumptions about h(x,u,∇u),we obtain\emph{a priori}estimates of positive solutions for the problem(0.1).Furthermore,we establish the existence of positive solutions by making use of these estimates and of the method of continuity.展开更多
We extend the fnite element method introduced by Lakkis and Pryer(SIAM J.Sci.Comput.33(2):786–801,2011)to approximate the solution of second-order elliptic problems in nonvariational form to incorporate the discontin...We extend the fnite element method introduced by Lakkis and Pryer(SIAM J.Sci.Comput.33(2):786–801,2011)to approximate the solution of second-order elliptic problems in nonvariational form to incorporate the discontinuous Galerkin(DG)framework.This is done by viewing the“fnite element Hessian”as an auxiliary variable in the formulation.Representing the fnite element Hessian in a discontinuous setting yields a linear system of the same size and having the same sparsity pattern of the compact DG methods for variational elliptic problems.Furthermore,the system matrix is very easy to assemble;thus,this approach greatly reduces the computational complexity of the discretisation compared to the continuous approach.We conduct a stability and consistency analysis making use of the unifed frameworkset out in Arnold et al.(SIAM J.Numer.Anal.39(5):1749–1779,2001/2002).We also give an a posteriori analysis of the method in the case where the problem has a strong solution.The analysis applies to any consistent representation of the fnite element Hessian,and thus is applicable to the previous works making use of continuous Galerkin approximations.Numerical evidence is presented showing that the method works well also in a more general setting.展开更多
COMPARED with the popular hyperspherical coordinate scheme, the HHGLF method proposed by Deng and others has the advantages of rapid hyperradial convergence, analytical solution and huge basis set calculation. However...COMPARED with the popular hyperspherical coordinate scheme, the HHGLF method proposed by Deng and others has the advantages of rapid hyperradial convergence, analytical solution and huge basis set calculation. However, the problem of slow convergence in the hyperangle part still exists as other hyperspherical harmonics (HH) methods do. To solve the prob-展开更多
基金supported by National Natural Science Foundation of China(11801167)Hunan Provincial Natural Science Foundation of China(2019JJ50387).
文摘In this paper,we will study the nonlocal and nonvariational elliptic problem{−(1+a||u||_(q)^(αq))Δu=|u|^(p−1)u+h(x,u,∇_(u))inΩ,u=0 on∂Ω,(0.1)(1)where a>0,α>0,1<q<2^(∗),p∈(0,2^(∗)−1)∖{1}andΩis a bounded smooth domain in R^(N)(N≥2).Under suitable assumptions about h(x,u,∇u),we obtain\emph{a priori}estimates of positive solutions for the problem(0.1).Furthermore,we establish the existence of positive solutions by making use of these estimates and of the method of continuity.
文摘We extend the fnite element method introduced by Lakkis and Pryer(SIAM J.Sci.Comput.33(2):786–801,2011)to approximate the solution of second-order elliptic problems in nonvariational form to incorporate the discontinuous Galerkin(DG)framework.This is done by viewing the“fnite element Hessian”as an auxiliary variable in the formulation.Representing the fnite element Hessian in a discontinuous setting yields a linear system of the same size and having the same sparsity pattern of the compact DG methods for variational elliptic problems.Furthermore,the system matrix is very easy to assemble;thus,this approach greatly reduces the computational complexity of the discretisation compared to the continuous approach.We conduct a stability and consistency analysis making use of the unifed frameworkset out in Arnold et al.(SIAM J.Numer.Anal.39(5):1749–1779,2001/2002).We also give an a posteriori analysis of the method in the case where the problem has a strong solution.The analysis applies to any consistent representation of the fnite element Hessian,and thus is applicable to the previous works making use of continuous Galerkin approximations.Numerical evidence is presented showing that the method works well also in a more general setting.
文摘COMPARED with the popular hyperspherical coordinate scheme, the HHGLF method proposed by Deng and others has the advantages of rapid hyperradial convergence, analytical solution and huge basis set calculation. However, the problem of slow convergence in the hyperangle part still exists as other hyperspherical harmonics (HH) methods do. To solve the prob-
