We consider interface problems for second order elliptic partial differential equations with Dirichlet boundary conditions. It is well known that the finite element discretization may fail to produce solutions converging with optimal rates unless the mesh fits with the discontinuity interface. We introduce a method based on piecewise linear finite elements on a non-matching grid enriched with a local correction on a sub-grid constructed along the interface. We prove that our method recovers the optimal convergence rates both in $H^1$ and in $L^2$ depending on the local regularity of the solution. Several numerical experiments confirm the theoretical results.
Local enrichment of finite elements for interface problems
CAVALLI, Fausto;GASTALDI, Lucia
2014-01-01
Abstract
We consider interface problems for second order elliptic partial differential equations with Dirichlet boundary conditions. It is well known that the finite element discretization may fail to produce solutions converging with optimal rates unless the mesh fits with the discontinuity interface. We introduce a method based on piecewise linear finite elements on a non-matching grid enriched with a local correction on a sub-grid constructed along the interface. We prove that our method recovers the optimal convergence rates both in $H^1$ and in $L^2$ depending on the local regularity of the solution. Several numerical experiments confirm the theoretical results.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.