Scattering for one dimensional perturbed Kirchhoff equations

Bernstein-Galerkin Approach for Perturbed Constant-Coefficient Differential Equations, One-Dimensional Analysis

A numerical approach for solving constant-coefficient differential equations whose solutions exhibit boundary layer structure is built by inserting Bernstein Partition of Unity into Galerkin variational weak form. Due to the reproduction capability of Bernstein basis, such implementation shows excellent accuracy at boundaries and is able to capture sharp gradients of the field variable by p-ref...

Superconvergence of Dg Method for One-dimensional Singularly Perturbed Problems

The convergence and superconvergence properties of the discontinuous Galerkin (DG) method for a singularly perturbed model problem in one-dimensional setting are studied. By applying the DG method with appropriately chosen numerical traces, the existence and uniqueness of the DG solution, the optimal order L2 error bounds, and 2p+1-order superconvergence of the numerical traces are established....

Convergence Analysis of Finite Element Solution of One-dimensional Singularly Perturbed Differential Equations on Equidistributing Meshes

In this paper convergence on equidistributing meshes is investigated. Equidistributing meshes, or more generally approximate equidistributing meshes, are constructed through the well-known equidistribution principle and a so-called adaptation (or monitor) function which is defined based on estimates on interpolation error for polynomial preserving operators. Detailed convergence analysis is giv...

Multiplicity Results for Perturbed Fourth-order Kirchhoff-type Problems

Numerical Methods for One-Dimensional Aggregation Equations

We focus in this work on the numerical discretization of the one dimensional aggregation equation ∂tρ + ∂x(vρ) = 0, v = a(W ′ ∗ ρ), in the attractive case. Finite time blow up of smooth initial data occurs for potential W having a Lipschitz singularity at the origin. A numerical discretization is proposed for which the convergence towards duality solutions of the aggregation equation is proved....