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Motivated by finite element spaces used for representation of temperature inthe compatible finite element approach for numerical weather prediction, weintroduce locally bounded transport schemes for (partially-)continuous finiteelement spaces. The underlying high-order transport scheme is constructed byinjecting the partially-continuous field into an embedding discontinuous finiteelement space, applying a stable upwind discontinuous Galerkin (DG) scheme, andprojecting back into the partially-continuous space; we call this an embeddedDG scheme. We prove that this scheme is stable in L2 provided that theunderlying upwind DG scheme is. We then provide a framework for applyinglimiters for embedded DG transport schemes. Standard DG limiters are appliedduring the underlying DG scheme. We introduce a new localised form ofelement-based flux-correction which we apply to limiting the projection backinto the partially-continuous space, so that the whole transport scheme isbounded. We provide details in the specific case of tensor-product finiteelement spaces on wedge elements that are discontinuous P1/Q1 in the horizontaland continuous P2 in the vertical. The framework is illustrated with numericaltests.

Embedded discontinuous Galerkin transport schemes with localised limiters