*u*=0 with non-linear boundary conditions. This non-linearboundary value problem is reformulated as a non-linear boundaryintegral equation, with

*u*on the boundary as the solution beingsought. The integral equation is solved numerically by usingthe collocation method, with piecewise quadratic functions usedas approximations to

*u*. Convergence results are given for thecases where (1) the original surface is used, and (2) the surfaceis approximated by piecewise quadratic interpolation. In addition,we define and analyze a two-grid iteration method for solvingthe non-linear system that arises from the discretization ofthe boundary integral equation. Numerical examples are given;and the paper concludes with a short discussion of the relativecost of different parts of the method. This work was supported in part by NSF grant DMS-9003287. 相似文献

Smooth integration is associated with a “pressure-vorticity” formulation which covers linear problems in elasticity and fluid mechanics. The solution presented herein is essentially the same as that reported in an earlier paper for regular elasticity. The constraint of incompressibility does not cause difficulties in the pressure-vorticity formulation.

The linear fluid mechanics problem formulated and solved in this paper covers Stokes' problem of a slow viscous flow, and has a wider interpretation. The translational inertia forces are incorporated in the linear problem, as in Euler's dynamic theory of inviscid flow. The centrifugal inertia forces are left for the non-linear problem. The linear problem is perceived as a step in solution of the non-linear problems. 相似文献

*Q*-curvature metrics on compact manifolds of arbitrary dimension under generic assumptions. The problem is equivalent to solving a

*n*th-order non-linear elliptic differential (or integral) equation with variational structure, where

*n*is the dimension of the manifold. Since the corresponding Euler functional is in general unbounded from above and below, we use critical point theory, jointly with a compactness result for the above equation. 相似文献

*L*

^{2}-sense. In particular, we derive a composite integral equation from Green's theorem which is uniquely solvable for all wave numbers. 相似文献

*L*

_{∞}(∂

*D*), ∂

*D*being the boundary of the exterior domain which we assume to be Lyapunov of index 1. For this exterior boundary value problem, we employ Green's theorem to derive a pair of boundary integral equations which have a unique simultaneous solution. We then show that this solution yields a solution of the original exterior boundary value problem. 相似文献

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^{3}. The obtained boundary integral equations for the unknown Cauchy data build a strong elliptic system of pseudodifferential equations which can therefore be used for numerical computations using Galerkin's procedure. We show existence, uniqueness and regularity of the solution of the integral equations. Especially we give the local "edge" behavior of the solution near the submanifold which divides the Dirichlet boundary from the Neumann boundary 相似文献