96 research outputs found
A subcell-enriched Galerkin method for advection problems
In this work, we introduce a generalization of the enriched Galerkin (EG)
method. The key feature of our scheme is an adaptive two-mesh approach that, in
addition to the standard enrichment of a conforming finite element
discretization via discontinuous degrees of freedom, allows to subdivide
selected (e.g. troubled) mesh cells in a non-conforming fashion and to use
further discontinuous enrichment on this finer submesh. We prove stability and
sharp a priori error estimates for a linear advection equation by using a
specially tailored projection and conducting some parts of a standard
convergence analysis for both meshes. By allowing an arbitrary degree of
enrichment on both, the coarse and the fine mesh (also including the case of no
enrichment), our analysis technique is very general in the sense that our
results cover the range from the standard continuous finite element method to
the standard discontinuous Galerkin (DG) method with (or without) local subcell
enrichment. Numerical experiments confirm our analytical results and indicate
good robustness of the proposed method
Bathymetry reconstruction using inverse shallow water models: Finite element discretization and regularization
In the present paper, we use modified shallow water equations (SWE) to reconstruct the bottom topography (also called bathymetry) of a flow domain without resorting to traditional inverse modeling techniques such as adjoint methods. The discretization in space is performed using a piecewise linear discontinuous Galerkin (DG) approximation of the free surface elevation and (linear) continuous finite elements for the bathymetry. Our approach guarantees compatibility of the discrete forward and inverse problems: for a given DG solution of the forward SWE problem, the underlying continuous bathymetry can be recovered exactly. To ensure well-posedness of the modified SWE and reduce sensitivity of the results to noisy data, a regularization term is added to the equation for the water height. A numerical study is performed to demonstrate the ability of the proposed method to recover bathymetry in a robust and accurate manner
Enriched Galerkin method for the shallow-water equations
This work presents an enriched Galerkin (EG) discretization for the two-dimensional shallow-water equations. The EG finite element spaces are obtained by extending the approximation spaces of the classical finite elements by discontinuous functions supported on elements. The simplest EG space is constructed by enriching the piecewise linear continuous Galerkin space with discontinuous, element-wise constant functions. Similarly to discontinuous Galerkin (DG) discretizations, the EG scheme is locally conservative, while, in multiple space dimensions, the EG space is significantly smaller than that of the DG method. This implies a lower number of degrees of freedom compared to the DG method. The EG discretization presented for the shallow-water equations is well-balanced, in the sense that it preserves lake-at-rest configurations. We evaluate the method’s robustness and accuracy using various analytical and realistic benchmarks and compare the results to those obtained using the DG method. Finally, we briefly discuss implementation aspects of the EG method within our MATLAB / GNU Octave framework FESTUNG
Frame-invariant directional vector limiters for discontinuous Galerkin methods
Second and higher order numerical approximations of conservation laws for vector fields call for the use of limiting techniques based on generalized monotonicity criteria. In this paper, we introduce a family of directional vertexbased slope limiters for tensor-valued gradients of formally second-order accurate piecewise-linear discontinuous Galerkin (DG) discretizations. The proposed methodology enforces local maximum principles for scalar products corresponding to projections of a vector field onto the unit vectors of a frame-invariant orthogonal basis. In particular, we consider anisotropic limiters based on singular value decompositions and the Gram-Schmidt orthogonalization procedure. The proposed extension to hyperbolic systems features a sequential limiting strategy and a global invariant domain fix. The pros and cons of different approaches to vector limiting are illustrated by the results of numerical studies for the two-dimensional shallow water equations and for the Euler equations of gas dynamics
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