Some error estimates for the lumped mass finite element method for a parabolic problem

We study the spatially semidiscrete lumped mass method for the model homogeneous heat equation with homogeneous Dirichlet boundary conditions. Improving earlier results we show that known optimal order smooth initial data error estimates for the standard Galerkin method carry over to the lumped mass method whereas nonsmooth initial data estimates require special assumptions on the triangulation. We also discuss the application to time discretization by the backward Euler and Crank-Nicolson methods

A First-Order Explicit-Implicit Splitting Method for a Convection-Diffusion Problem

We analyze a second-order in space, first-order in time accurate finite difference method for a spatially periodic convection-diffusion problem. This method is a time stepping method based on the first-order Lie splitting of the spatially semidiscrete solution. In each time step, on an interval of length k, of this solution, the method uses the backward Euler method for the diffusion part, and then applies a stabilized explicit forward Euler approximation on m >= 1 intervals of length k/m for the convection part. With h the mesh width in space, this results in an error bound of the form C(0)h(2) + C(m)k for appropriately smooth solutions, where C-m <= C\u27 + C-\u27\u27/m. This work complements the earlier study [V. Thomee and A. S. Vasudeva Murthy, An explicit- implicit splitting method for a convection-diffusion problem, Comput. Methods Appl. Math. 19 (2019), no. 2, 283-293] based on the second-order Strang splitting

An Explicit-Implicit Splitting Method for a Convection-Diffusion Problem

We analyze a second-order accurate finite difference method for a spatially periodic convection-diffusion problem. The method is a time stepping method based on the Strang splitting of the spatially semidiscrete solution, in which the diffusion part uses the Crank-Nicolson method and the convection part the explicit forward Euler approximation on a shorter time interval. When the diffusion coefficient is small, the forward Euler method may be used also for the diffusion term

A finite element splitting method for a convection-diffusion problem

For a spatially periodic convection-diffusion problem, we analyze a time stepping method based on Lie splitting of a spatially semidiscrete finite element solution on time steps of length k, using the backward Euler method for the diffusion part and a stabilized explicit forward Euler approximation on m ≥ 1 intervals of length k/m for the convection part. This complements earlier work on time splitting of the problem in a finite difference context

From Finite Differences to Finite Elements - A short history of numerical analysis of partial differential equations

This is an account of the history of numerical analysis of partial differential equations, starting with the 1928 paper of Courant, Friedrichs, and Lewy, and proceeding with the development of first finite difference and then finite element methods. The emphasis is on mathematical aspects such as stability and convergence analysis

Partial differential equations with numerical methods

The book is suitable for advanced undergraduate and beginning graduate students of applied mathematics and engineering. The main theme is the integration of the theory of linear PDEs and the numerical solution of such equations. For each type of PDE, elliptic, parabolic, and hyperbolic, the text contains one chapter on the mathematical theory of the differential equation, followed by one chapter on finite difference methods and one on finite element methods. As preparation, the two-point boundary value problem and the initial-value problem for ODEs are discussed in separate chapters. There is also one chapter on the elliptic eigenvalue problem and eigenfunction expansion. The presentation does not presume a deep knowledge of mathematical and functional analysis. Some background on linear functional analysis and Sobolev spaces, and also on numerical linear algebra, is reviewed in two appendices