In this paper, we propose a derivative-free projection method for solving nonlinear equations, which combines the prediction-correction technique with the hyperplane projection technique. Its search direction is a modified spectral gradient direction. Under appropriate assumptions, the search direction satisfies both the sufficient descent property and the trust region property, and the global convergence of the proposed method is established. Furthermore, with the help of the error bound condition, we prove the R-linear convergence rate of the proposed method. Numerical experiments demonstrate that the prediction-correction technique significantly improves the computational performance.
We present the first comprehensive pointwise a posteriori error analysis for hybrid high-order (HHO) methods applied to the Poisson problem on general polytopal meshes in 2D and 3D. The method employs scalar-valued polynomials across mesh faces and elements as degrees of freedom. The core of the analysis is an element-wise local discrete gradient reconstruction operator. To connect the non-conforming HHO framework to the conforming setting, we construct an enriching operator of Oswald-averaging type, following the approach of Demlow and Georgoulis [1], which allows us to decompose the error into conforming and non-conforming parts. The article thoroughly discusses the pointwise efficiency and reliability of the proposed a posteriori error estimator. Finally, numerical experiments illustrate the theoretical findings and demonstrate the method’s flexibility and computational efficiency compared to the discontinuous Galerkin approach.
We study fully discrete local discontinuous Galerkin (LDG) methods which adopt the implicit-explicit (IMEX) backward difference formulas (BDF) from first to fifth order in time discretization, for one-dimensional linear convection-diffusion problems under either Dirichlet or Neumann boundary conditions. The unconditional stability of the LDG-IMEX-BDF schemes up to fifth order in time is investigated, by exploiting the multiplier technique as well as the technique to deal with derivatives and jumps of the numerical solution. The Gauss-Radau (GR) projection is used to derive optimal error estimates in the Dirichlet case, and a hybrid GR projection is proposed to achieve optimal error estimates in the Neumann case. Numerical experiments for both linear and nonlinear convection-diffusion equations confirm the optimal accuracy of the proposed schemes.
