2 research outputs found
On Approximating Discontinuous Solutions of PDEs by Adaptive Finite Elements
For singularly perturbed problems with a small diffusion, when the transient
layer is very sharp and the computational mesh is relatively coarse, the
solution can be viewed as discontinuous. For both linear and nonlinear
hyperbolic partial differential equations, the solution can be discontinuous.
When finite element methods with piecewise polynomials are used to approximate
these discontinuous solutions, numerical solutions often overshoot near a
discontinuity. Can this be resolved by adaptive mesh refinements?
In this paper, for a simple discontinuous function, we explicitly compute its
continuous and discontinuous piecewise constant or linear projections on
discontinuity matched or non-matched meshes. For the simple
discontinuity-aligned mesh case, piecewise discontinuous approximations are
always good. For the general non-matched case, we explain that the piecewise
discontinuous constant approximation combined with adaptive mesh refinements is
the best choice to achieve accuracy without overshooting. For discontinuous
piecewise linear approximations, non-trivial overshootings will be observed
unless the mesh is matched with discontinuity. For continuous piecewise linear
approximations, the computation is based on a "far away assumption", and
non-trivial overshootings will always be observed under regular meshes. We
calculate the explicit overshooting values for several typical cases.
Several numerical tests are preformed for a singularly-perturbed
reaction-diffusion equation and linear hyperbolic equations to verify our
findings in the paper.Comment: 23 page