Topic
Finite difference method
About: Finite difference method is a research topic. Over the lifetime, 21603 publications have been published within this topic receiving 468852 citations. The topic is also known as: Finite-difference methods & FDM.
Papers published on a yearly basis
1 / 2
Papers
TL;DR: A new approach to error control and mesh adaptivity is described for the discretization of optimal control problems governed by elliptic partial differential equations and the computed state and costate variables can be used as sensitivity factors multiplying the local cell-residuals in the error estimators.
Abstract: A new approach to error control and mesh adaptivity is described for the discretization of optimal control problems governed by elliptic partial differential equations. The Lagrangian formalism yields the first-order necessary optimality condition in form of an indefinite boundary value problem which is approximated by an adaptive Galerkin finite element method. The mesh design in the resulting reduced models is controlled by residual-based a posteriori error estimates. These are derived by duality arguments employing the cost functional of the optimization problem for controlling the discretization error. In this case, the computed state and costate variables can be used as sensitivity factors multiplying the local cell-residuals in the error estimators. This results in a generic and simple algorithm for mesh adaptation within the optimization process. This method is developed and tested for simple boundary control problems in semiconductor models.
342 citations
TL;DR: In this paper, the authors consider linear, selfadjoint, elliptic problems with Neumann boundary conditions in rectangular domains and demonstrate that with sufficiently smooth data, the discrete $L^2 $-norm errors for tensor product block-centered finite differences in both the approximate solution and its first derivatives are second-order for all nonuniform grids.
Abstract: We consider linear, selfadjoint, elliptic problems with Neumann boundary conditions in rectangular domains. We demonstrate that with sufficiently smooth data, the discrete $L^2 $-norm errors for tensor product block-centered finite differences in both the approximate solution and its first derivatives are second-order for all nonuniform grids. Extensions to nonselfadjoint and parabolic problems are discussed.
341 citations
Book•
1 Apr 1990
TL;DR: This book presents the simplest description of continuous fluid flow, transport as a combination of addiction and diffusion, and solution methods for unsteady free surface flows.
Abstract: 1. Introduction. 2. The simplest description of continuous fluid flow. 3. The finite difference method. 4. Diffusion problems. 5. Transport as a combination of addiction and diffusion. 6. Descriptions of unsteady flows. 7. Solution methods for unsteady free surface flows. 8. Equilibrium methods. 9. Computational fluid dynamics of turbulence. 10. An introduction to some other numerical methods. Index.
341 citations
TL;DR: In this article, a numerical scheme to solve the one-dimensional nonlinear Klein-Gordon equation with quadratic and cubic nonlinearity is proposed, which uses the collocation points and approximates the solution using Thin Plate Splines (TPS) radial basis functions (RBF).
340 citations
TL;DR: In this paper, a nonconforming Virtual Element Method (VEM) was proposed for the approximation of second order elliptic problems. But the method is not suitable for finite element methods.
Abstract: We introduce the nonconforming Virtual Element Method (VEM) for the approximation of second order elliptic problems. We present the construction of the new element in two and three dimensions, highlighting the main differences with the conforming VEM and the classical nonconforming finite element methods. We provide the error analysis and establish the equivalence with a family of mimetic finite difference methods. Numerical experiments verify the theory and validate the performance of the proposed method.
340 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 53 |
| 2024 | 136 |
| 2023 | 170 |
| 2022 | 357 |
| 2021 | 733 |
| 2020 | 690 |