Residual based error estimate and quasi-interpolation on polygonal meshes for high order BEM-based FEM

2019

ESAIM: M2AN

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New interpolation and quasi-interpolation operators of Clément-and Scott-Zhang-type are analyzed on anisotropic polygonal and polyhedral meshes. Since no reference element is available, an appropriate linear mapping to a reference configuration plays a crucial role. A priori error estimates are derived respecting the anisotropy of the discretization. Finally, the found estimates are employed to propose an adaptive mesh refinement based on bisection which leads to highly anisotropic and adapted discretizations with general element shapes in two-and three-dimensions.2010 Mathematics Subject Classification. 65D05, 65N15, 65N30, 65N50.

Section: 2mentioning

confidence: 99%

“…Quasi-interpolation operators on anisotropic simplicial meshes can be found in [2,18], for example. Clément-type interpolation operators on polygonal meshes have been studied in [26,28].…”

Section: Quasi-interpolation Of Non-smooth Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Anisotropic polygonal and polyhedral discretizations in finite element analysis

2019

ESAIM: M2AN

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“…Afterwards, the meshes T (F ) of level ≥ 1 are defined recursively by splitting each triangle of the previous level into four similar triangles by connecting its edge midpoints. The initial mesh T 0 (F ) is shape-regular in the sense of Ciarlet, where the regularity parameters only depend on the regularity parameters of the polyhedral mesh, see [44]. Consequently, all successive refinements T (F ) are also regular.…”

Section: Approximation Of the Skeletal Basis Functionsmentioning

confidence: 99%

“…In [32], a new construction of the approximation space was proposed, which employs the polygonal faces of the polyhedral elements. Residual-type a posteriori discretization error estimates were derived in [39] and extended in [44,45]. These a posteriori discretization error estimates can be used to derive adaptive versions of the BEM-based FEM, see also the PhD thesis by S. Weißer [40].…”

mentioning

confidence: 99%

Convection-adapted BEM-based FEM

Hofreither

Langer

2016

Z. Angew. Math. Mech.

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We present a new discretization method for homogeneous convectiondiffusion-reaction boundary value problems in 3D that is a non-standard finite element method with PDE-harmonic shape functions on polyhedral elements. The element stiffness matrices are constructed by means of local boundary element techniques. Our method, which we refer to as a BEM-based FEM, can therefore be considered a local Trefftz method with element-wise (locally) PDE-harmonic shape functions. The Dirichlet boundary data for these shape functions is chosen according to a convection-adapted procedure which solves projections of the PDE onto the edges and faces of the elements. This improves the stability of the discretization method for convection-dominated problems both when compared to a standard FEM and to previous BEM-based FEM approaches, as we demonstrate in several numerical experiments.

Gradient Recovery for the BEM‐based FEM and VEM

Seibel

Proc Appl Math and Mech

2019

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In this article, we propose new gradient recovery schemes for the BEM-based Finite Element Method (BEM-based FEM) and Virtual Element Method (VEM). Supporting general polytopal meshes, the BEM-based FEM and VEM are highly flexible and efficient tools for the numerical solution of boundary value problems in two and three dimensions. We construct the recovered gradient from the gradient of the finite element approximation via local averaging. For the BEM-based FEM, we show that, under certain requirements on the mesh, superconvergence of the recovered gradient is achieved, which means that it converges to the true gradient at a higher rate than the untreated gradient. Moreover, we propose a simple and very efficient a posteriori error estimator, which measures the difference between the unprocessed and recovered gradient as an error indicator. Since the BEM-based FEM and VEM are specifically suited for adaptive refinement, the resulting adaptive algorithms perform very well in numerical examples.