The 𝐾-operator and the Galerkin method for strongly elliptic equations on smooth curves: local estimates

Abstract

Abstract. Superconvergence in the L2-norm for the Galerkin approximation of the integral equation Lu = f is studied, where I is a strongly elliptic pseudodifferential operator on a smooth, closed or open curve. Let Uf, be the Galerkin approximation to u . By using the ^-operator, an operator that averages the values of uh , we will construct a better approximation than uh itself. That better approximation is a legacy of the highest order of convergence in negative norms. For Symm's equation on a slit the same order of convergence can be recovered if the mesh is suitably graded.

Local convergence of the boundary element method on polyhedral domains

“…Numerical examples show the sharpness of our analysis. As discussed in Remark 2.5 below, our results improve [ 21 , 27 , 28 ] as estimates in (for Symm’s equation) and (for the hyper-singular equation) are obtained there from local energy norm estimates with the aid of inverse estimates, thereby leading to a loss of . In contrast, we avoid using an inverse inequality to go from the energy norm to a stronger norm.…”

“…Significantly fewer works study the local behavior of the BEM. The case of smooth two dimensional curves is treated in [ 5 , 21 , 28 ], in [ 27 ] three dimensional screen problems are studied, and [ 14 ] discusses local error estimates on polygons. [ 19 , 20 ] provide estimates in the -norm on smooth domains.…”

“…However, for the case of piecewise smooth geometries such as polygonal and polyhedral domains, sharp local error estimates that exploit the maximal (local) regularity of the solution are not available. Moreover, the analyses of [ 14 , 21 , 27 , 28 ] are tailored to the energy norm and do not provide optimal local estimates in stronger norms, whereas [ 5 ] imposes additional global regularity.…”