High-Order Local Rate of Convergence By Mesh-Refinement in the Finite Element Method

We seek approximations of the solution u of the Neumann problem for the equation Lu = f in Q with special emphasis on high-order accuracy at a given point x0 e Q. Here ß is a bounded domain in R (N > 2) with smooth boundary, and L is a second-order, uniformly elliptic, differential operator with smooth coefficients. An approximate solution uh is determined by the standard Galerkin method in a space of continuous piecewise polynomials of degree at most r 1 on a partition Ah(x0, a) of Í2. Here h is a global mesh-size parameter, and a is the degree of a certain systematic refinement of the mesh around the given point x0, where larger a's mean finer mesh, and a = 0 corresponds to the quasi-uniform case with no refinement. It is proved that, for suitable (sufficiently large) a's the high-order error estimate (u uh)(x0) = 0(h2r~2) holds. A corresponding estimate with the same order of convergence is obtained for the first-order derivatives of u uh. These estimates are sharp in the sense that the required degree of refinement in each case is essentially the same as is needed for the local approximation to this order near x0. For the estimates to hold, it is sufficient that the exact solution u have derivatives to the rth order which are bounded close to x0 and square integrable in the rest of Q. The proof of this uses high-order negative-norm estimates of u uh. The number of elements in the considered partitions is of the same order as in the corresponding quasi-uniform ones. Applications of the results to other types of boundary value problems are indicated. 0. Introduction. Let Q be a bounded domain in RN, N > 2, with smooth boundary 9Í2 and consider the Neumann problem to find u such that N a / a \ N o d / du \ v-^ o« (0.1) Lu= E ö +Iû +f,ttÄ/ infl, i,i=i j \ ' ' i = i ' (0.2) ir-E «JrS = 0 on9ßHere L is assumed to be uniformly elliptic with smooth coefficients, n = (n,) and nc denote the exterior normal and conormal to 3Í2, respectively. Assume also that the bilinear form A(v,w) = )a | J>|y_-_ + £ a—w + avwj dx Received September 22, 1983; revised July 23, 1984. 1980 Mathematics Subject Classification. Primary 65N30.