Approximations of Very Weak Solutions to Boundary-Value Problems

Standard weak solutions to the Poisson problem on a bounded domain have square-integrable derivatives, which limits the admissible regularity of inhomogeneous data. The concept of solution may be further weakened in order to define solutions when data is rough, such as for inhomogeneous Dirichlet data that is only square-integrable over the boundary. Such very weak solutions satisfy a nonstandard variational form (u,v) = G(v). A Galerkin approximation combined with an approximation of the right-hand side G defines a finite-element approximation of the very weak solution. Applying conforming linear elements leads to a discrete solution equivalent to the text-book finite-element solutionto the Poisson problem in which the boundary data is approximated by L2-projections. The L2 convergence rate of the discrete solution is O(hs) for some $s\in(0,1/2)$ that depends on the shape of the domain, assuming a polygonal (two-dimensional) or polyhedral (three-dimensional) domain without slits and (only) square-integrable boundary data.