2020
DOI: 10.1515/cmam-2020-0042
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Adaptive Space-Time Finite Element Methods for Non-autonomous Parabolic Problems with Distributional Sources

Abstract: We consider locally stabilized, conforming finite element schemes on completely unstructured simplicial space-time meshes for the numerical solution of parabolic initial-boundary value problems with variable coefficients that are possibly discontinuous in space and time. Distributional sources are also admitted. Discontinuous coefficients, non-smooth boundaries, changing boundary conditions, non-smooth or incompatible initial conditions, and non-smooth right-hand sides can lead to non-smooth solutions. We pres… Show more

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Cited by 16 publications
(12 citation statements)
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“…In this section, we will briefly describe the space-time finite element method based on localized time-upwind stabilizations; for details of the construction and analysis, we refer to our previous work [8,10,9]. Let T h be a shape regular decomposition of the space-time cylinder Q, i.e., Q = K∈T h K, and K ∩K = ∅ for all K and K from T h with K = K ; see, e.g., [2] for more details.…”
Section: Space-time Finite Element Methodsmentioning
confidence: 99%
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“…In this section, we will briefly describe the space-time finite element method based on localized time-upwind stabilizations; for details of the construction and analysis, we refer to our previous work [8,10,9]. Let T h be a shape regular decomposition of the space-time cylinder Q, i.e., Q = K∈T h K, and K ∩K = ∅ for all K and K from T h with K = K ; see, e.g., [2] for more details.…”
Section: Space-time Finite Element Methodsmentioning
confidence: 99%
“…Using Galerkin orthogonality (subtracting ( 4) from ( 2)), and coercivity and extended boundedness of the bilinear form (3), we can show the following Céa-like best approximation estimate; see [8,10,9] for the proofs.…”
Section: Space-time Finite Element Methodsmentioning
confidence: 99%
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“…Adaptive mesh refinement can produce space-time discretizations that yield accurate solutions with relatively few degrees of freedom. For example, Langer & Schafelner [4] compared space-time FEMs using uniformly and adaptively refined meshes; they found that obtaining the same approximation error for their tests required more degrees of freedom on the uniform meshes by one to two orders of magnitude.…”
Section: Introductionmentioning
confidence: 99%