Yanping Lin

On the Accuracy of the Finite Volume Element Method Based on Piecewise Linear Polynomials

We present a general error estimation framework for a finite volume element (FVE) method based on linear polynomials for solving second-order elliptic boundary value problems. This framework treats the FVE method as a perturbation of the Galerkin finite element method and reveals that regularities in both the exact solution and the source term can affect the accuracy of FVE methods. In particular, the error estimates and counterexamples in this paper will confirm that the FVE method cannot have the standard O(h2) convergence rate in the L2 norm when the source term has the minimum regularity, only being in L2, even if the exact solution is in H2.

An inverse problem of finding a parameter in a semi-linear heat equation

We consider the following inverse problem of finding the pair (u, p) which satisfies the following: u1 = uxx + p(t)u + F(x, t, u, ux, p(t), 0 < x < 1, 0 < t ⩽ T; u(x,0) = u0(x), 0 < x < 1, ux(0, t) = f(t), ux(1, t) = g(t), 0 < t ⩽ T; and ∝01ψ(x,t) u(x, t)dx = E(t), 0 < t ⩽ T; where u0, f, g, F, ψ, and E are known functions. The existence, uniqueness, regularity, and the continuous dependence of the solution upon the data are demonstrated.

Ritz-Volterra projections to finite-element spaces and applications to integrodifferential and related equations

The object of this paper is to investigate the convergence of finite-element approximations to solutions of parabolic and hyperbolic integrodifferential equations, and also of equations of Sobolev and viscoelasticity type. The concept of Ritz–Volterra projection will be seen to unify much of the analysis for the different types of problems. Optimal order error estimates are obtained in

A finite-difference solution to an inverse problem for determining a control function in a parabolic partial differential equation

L_p

Finite volume element approximations of nonlocal reactive flows in porous media

for

A priori L 2 error estimates for finite-element methods for nonlinear diffusion equations with memory

2 \leqq p < \infty

The immersed finite volume element methods for the elliptic interface problems

, and almost optimal order pointwise results given.

Sharp L 2 -Error Estimates and Superconvergence of Mixed Finite Element Methods for Non-Fickian Flows in Porous Media

A finite-difference solution is demonstrated for an inverse problem of determining a control function p(t) in the parabolic partial differential equation ut=uxx+pu+f(x,t), 0

The solution of the diffusion equation in two space variables subject to the specification of mass

In this article, we study finite volume element approximations for two-dimensional parabolic integro-differential equations, arising in the modeling of nonlocal reactive flows in porous media. These types of flows are also called NonFickian flows and exhibit mixing length growth. For simplicity, we consider only linear finite volume element methods, although higher-order volume elements can be considered as well under this framework. It is proved that the finite volume element approximations derived are convergent with optimal order in H1- and L2-norm and are superconvergent in a discrete H1-norm. By examining the relationship between finite volume element and finite element approximations, we prove convergence in L∞- and W1,∞-norms. These results are also new for finite volume element methods for elliptic and parabolic equations.

Partially Penalized Immersed Finite Element Methods For Elliptic Interface Problems

This paper studies finite-element approximations to the solutions of the nonlinear diffusion equations with memory. An elliptic projection with memory associated with our equations is defined and then used in the derivations of optimal error estimates for semidiscrete and Crank–Nicolson finite-element approximations.