# Error estimation for interface problems

## Description

Here we present an error estimation method developed for the numerical solutions of interface problems. The goal is to approximate the error of a given numerical approximation to the true solution. As opposed to classic a priori error estimates which may seek to bound the norm of the error by some expression involving a discretization parameter, the focus of this work will be to produce numerical values of the errors associated with a particular solution. The spatial distribution of these errors is also available. This type of estimate is useful in settings where simulations are used as replacements for experiments. In such settings it is important to be able to understand, and, if possible, quantify such errors We begin by describing the Method of Nearby Problems, [1, 2, 3], which is motivated by defect correction methods [4]. The general idea of the Method of Nearby Problems is to construct a problem for which an analytic solution is known and, at the same time, is a small perturbation of the original problem of interest. With this idea in mind, we move forward to describe a modification of the original Method of Nearby Problems that we will study here Although our ultimate goal is to apply the error estimates to numerical solutions of interface problems, we first examine their effectiveness on continuous elliptic problems. We will construct these estimates and study their convergence to the true error as the mesh is refined. We also provide analysis of the error estimate for a one-dimensional example The next step will be to examine the application and performance of the estimates to numerical solutions of interface problems. Our model problem will be an elliptic boundary value problem in which the coefficients are discontinuous across an internal boundary. We will obtain numerical solutions of these problems using the Immersed Interface Method [5, 6]. The Immersed Interface Method solutions preserve the discontinuities and irregularities of the solution at the interface We will next examine solutions obtained using regularization methods. Motivated by the Immersed Boundary Method [7], we regularize the interface problem and then use numerical methods which apply to smooth problems. Here we must analyze both the error due to regularization, as well as discretization error. We will apply this methodology to the discontinuous boundary value problem and provide an analysis of the regularization error for a representative one-dimensional problem Finally, we discuss future directions such as generalizing the method to include Immersed Interface solutions of problems with singular sources. We also discuss requirements of error estimates for transient problems