Introduction To Boundary Integral Equation Methods
Introduction to boundaryintegral equation methodsArnold D. KimBIERS MeetingSeptember 25, 2018
What are boundary integral equations? We can reformulate boundary value problems for PDEs in adomain as integral equations on the boundary of that domain. We typically use them for linear, elliptic, and homogeneousPDEs, but not always. Boundary integral equation methods refer to the numericalsolution of these integral equations.
Why study boundary integral equations? Since we only solve a problem on the boundary, there is anoverall reduction in dimension by one. It is “meshless” so it can compute solutions in complicateddomains. Numerics rule of thumb: “It’s better to integrate than todifferentiate.”
What are the challenges? The boundary integral formulation is initially more abstract/lessintuitive. Numerical solution of PDEs yield sparse matrices, whilenumerical solutions of boundary integral equations yield densematrices. Technical challenges regarding error analysis.
Trying to build some intuitionHow can we reduce the solution of a PDE in a domain to an BIE on itsboundary?Consider the two-point boundary value problem in one dimension.The solution we seek must satisfy (1) the DE and (2) the BCs.
Trying to build some intuitionWe write the solution as a linear combination of two functions thatsatisfy the DE.By requiring that this function satisfies the BCs, we find that thecoefficients must satisfy
Trying to build some intuition For a boundary value problem, we need to satisfy thedifferential equation and the boundary conditions. Suppose we seek the solution as a superposition of solutions ofthe differential equation. Then that solution automaticallysatisfies the differential equation. All that is left is to satisfy the boundary conditions.
Extending this ideaSuppose we would like to solve the interior, Dirichlet boundaryvalue problem in two/three dimensions.Can we write the solution as a “superposition” of functions thatsatisfy Laplace’s equation, and then require that thissuperposition satisfies the boundary conditions?
Skipping some analysis We write the solution as the continuous superposition,It can be shown thatdensitykernel
Boundary integral equationNow that we have a function that satisfies the PDE, we require itto satisfy the BC. However, we note thatIt follows that the density satisfies
Boundary integral equation method1. Solve boundary integral equation,2. Evaluate the solution,
What have we skipped? We have introduced a solution in terms of the fundamental solutionof the PDE (Green’s functions, Green’s theorems, etc.). This solution has a specific jump when evaluated on the boundary(Gauss’ theorem). How to compute the numerical solution of the boundary integralequation and how to compute the numerical evaluation of thesolution? Applications where this method is useful.
What are my interests in BIE methods?With BIE methods, we have an explicit expression for thesolution, e.g.That means that this representation contains all of the physicalbehavior of a system.Can we use this to extract valuable insight into complex problemsthrough asymptotic analysis of this expression?
Any Questions?
What are boundary integral equations? We can reformulate boundary value problems for PDEs in a domain as integral equations on the boundary of that domain. We typically use them for linear, elliptic, and homogeneous PDEs, but not always. Boundary integral equation methods refer to the numeric
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