A local discontinuous Galerkin gradient discretization method for linear and quasilinear elliptic equations
For researchers in numerical PDEs, this provides a locally adaptive scheme with theoretical guarantees for elliptic problems, though the method is incremental.
The paper introduces a local weighted discontinuous Galerkin gradient discretization method for linear and quasilinear elliptic equations, proving convergence under minimal regularity and showing that errors from artificial boundary conditions are higher-order and locally dependent on solution regularity. Numerical experiments demonstrate accuracy improvements over the non-local approach.
A local weighted discontinuous Galerkin gradient discretization method for solving elliptic equations is introduced. The local scheme is based on a coarse grid and successively improves the solution solving a sequence of local elliptic problems in high gradient regions. Using the gradient discretization framework we prove convergence of the scheme for linear and quasilinear equations under minimal regularity assumptions. The error due to artificial boundary conditions is also analyzed, shown to be of higher order and shown to depend only locally on the regularity of the solution. Numerical experiments illustrate our theoretical findings and the local method's accuracy is compared against the non local approach