Discontinuous Galerkin methods for problems with Dirac delta source

Abstract

Abstract. In this article we study discontinuous Galerkin finite element discretizations of linear second-order elliptic partial differential equations with Dirac delta right-hand side. In particular, assuming that the underlying computational mesh is quasi-uniform, we derive an a priori bound on the error measured in terms of the L 2 -norm. Additionally, we develop residual-based a posteriori error estimators that can be used within an adaptive mesh refinement framework. Numerical examples for the symmetric interior penalty scheme are presented which confirm the theoretical results.Mathematics Subject Classification. 65N30.

An Exactly Mass Conserving and Pointwise Divergence Free Velocity Method: Application to Compositional Buoyancy Driven Flow Problems in Geodynamics

“…Given an SIME ET AL. (Scott, 1973;Houston & Wihler, 2012). Does not provide a field representation ϕ h (x, t) required by the viscosity model Dirac delta projection L 2 projection of tracer data to an FE space.…”

“…In general, however, the integration of Dirac delta functions does not satisfy the smoothness, or regularity, requirements for finite elements. This causes algorithms that depend on it to typically experience reduced error convergence rates even when using nonconforming methods (e.g., Houston & Wihler, 2012; Scott, 1973). When the composition field is used to drive or influence the flow, for example through the buoyancy and viscosity terms, respectively, this may in turn lead to suboptimal convergence of the velocity approximation.…”

“…Does not satisfy regularity requirements of the FE scheme. Will reduce convergence rate of the error of the velocity FE approximation to (Scott, 1973; Houston & Wihler, 2012). Does not provide a field representation ϕ h ( x , t ) required by the viscosity model…”

A hybridizable discontinuous Galerkin method for second order elliptic equations with Dirac delta source

“…)). In 2012, Houston and Wihler [26] studied the discontinuous Galerkin (DG) methods for problem (1) with d = 2. The convergence rate O(h) for L 2 -error was proved under a constraint that x 0 lies in the interior of an element.…”

An exactly mass conserving and pointwise divergence free velocity method: application to compositional buoyancy driven flow problems in geodynamics

“…In general, however, the integration of Dirac delta functions does not satisfy the smoothness, or regularity, requirements for finite elements. This causes algorithms that depend on it to typically experience reduced error convergence rates even when using nonconforming methods (e.g., Houston & Wihler, 2012;Scott, 1973). When the composition field is used to drive or influence the flow, for example through the buoyancy and viscosity terms, respectively, this may in turn lead to suboptimal convergence of the velocity approximation.…”