A posteriori error estimators in the finite element method.
A posteriori estimates for the Cahn–Hilliard equation with obstacle free energy
We derive a posteriori estimates for a discretization in space of the standard Cahn–Hilliard equation with a double obstacle free energy. The derived estimates are robust and efficient, and in practice are combined with a heuristic time step adaptation. We present numerical experiments in two and three space dimensions and compare our method with an existing heuristic spatial mesh adaptation algorithm.
A posteriori estimations of approximate solutions for some types of boundary value problems
A posteriori upper and lower error bound of the high-order discontinuous Galerkin method for the heat conduction equation
We deal with the numerical solution of the nonstationary heat conduction equation with mixed Dirichlet/Neumann boundary conditions. The backward Euler method is employed for the time discretization and the interior penalty discontinuous Galerkin method for the space discretization. Assuming shape regularity, local quasi-uniformity, and transition conditions, we derive both a posteriori upper and lower error bounds. The analysis is based on the Helmholtz decomposition, the averaging interpolation...
A Potential Method for the Biharmonic Equation.
A power series method for computing singular solutions to nonlinear analytic systems.
A Practical Finite Element Approximation of a Semi-Definite Neumann Problem on a Curved Domain.
A preconditioner for a FETI-DP method for mortar element discretization of a 4th order problem in 2D.
A preconditioner for the FETI-DP method for mortar-type Crouzeix-Raviart element discretization
In this paper, we consider mortar-type Crouzeix-Raviart element discretizations for second order elliptic problems with discontinuous coefficients. A preconditioner for the FETI-DP method is proposed. We prove that the condition number of the preconditioned operator is bounded by , where and are mesh sizes. Finally, numerical tests are presented to verify the theoretical results.
A Predictor-Corrector Technique for Constrained Least-Squares Regularization.
A Primal Hybrid Finite Element Method for Quasilinear Second Order Elliptic Problems.
A primal-dual algorithm for a constrained Fermat-Weber problem involving mixed norms
A primal-dual integral method in global optimization
Using the Fenchel conjugate of Phú’s Volume function F of a given essentially bounded measurable function f defined on the bounded box D ⊂ Rⁿ, the integral method of Chew and Zheng for global optimization is modified to a superlinearly convergent method with respect to the level sequence. Numerical results are given for low dimensional functions with a strict global essential supremum.
A principle of dehomogenization for eigenvalue problems (Preliminary communication)
A priori and a posteriori error bounds for a nonconforming linear finite element approximation of a non-newtonian flow
A priori bounds for a semilinear wave equation
A priori convergence of the greedy algorithm for the parametrized reduced basis method
The convergence and efficiency of the reduced basis method used for the approximation of the solutions to a class of problems written as a parametrized PDE depends heavily on the choice of the elements that constitute the “reduced basis”. The purpose of this paper is to analyze the a priori convergence for one of the approaches used for the selection of these elements, the greedy algorithm. Under natural hypothesis on the set of all solutions to the problem obtained when the parameter varies, we...
A priori convergence of the Greedy algorithm for the parametrized reduced basis method
The convergence and efficiency of the reduced basis method used for the approximation of the solutions to a class of problems written as a parametrized PDE depends heavily on the choice of the elements that constitute the “reduced basis”. The purpose of this paper is to analyze the a priori convergence for one of the approaches used for the selection of these elements, the greedy algorithm. Under natural hypothesis on the set of all solutions to the problem obtained when the parameter varies, we...
A priori convergence of the Greedy algorithm for the parametrized reduced basis method
The convergence and efficiency of the reduced basis method used for the approximation of the solutions to a class of problems written as a parametrized PDE depends heavily on the choice of the elements that constitute the “reduced basis”. The purpose of this paper is to analyze the a priori convergence for one of the approaches used for the selection of these elements, the greedy algorithm. Under natural hypothesis on the set of all solutions to the problem obtained when the parameter varies, we...
A priori diffusion-uniform error estimates for singularly perturbed problems: Midpoint-DG discretization
We deal with a nonstationary semilinear singularly perturbed convection–diffusion problem. We discretize this problem by discontinuous Galerkin method in space and by midpoint rule in time. We present diffusion–uniform error estimates with sketches of proofs.