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 present new a priori and a posteriori error estimates for low-regularity solutions. In order to avoid reduced rates of convergence that appear when performing uniform mesh refinement, we also consider adaptive refinement procedures based on residual a posteriori error indicators and functional a posteriori error estimators. The huge system of space-time finite element equations is then solved by means of GMRES preconditioned by space-time algebraic multigrid. In particular, in the 4d space-time case, simultaneous space-time parallelization can considerably reduce the computational time. We present and discuss numerical results for several examples possessing different regularity features.

Space-time hexahedral finite element methods for parabolic evolution problems

“…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.…”

“…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.…”

“…In this paper, we follow our preceding papers [8,10,9], and construct locally stabilized, conforming space-time finite element schemes for solving the IBVP (1), but on hexahedral meshes that are more suited for anisotropic refinement than simplicial meshes used in [8,10,9]. We mention that SUPG/SD and Galerkin/least-squares stabilizations of time-slice finite element schemes for solving transient problems were already used in early papers; see, e.g., [7] and [6].…”

“…We mention that SUPG/SD and Galerkin/least-squares stabilizations of time-slice finite element schemes for solving transient problems were already used in early papers; see, e.g., [7] and [6]. Section 2 recalls the construction of locally stabilized space-time finite element schemes, the properties of the corresponding discrete bilinear form, and the a priori discretization error estimates from [8,10,9]. In Section 3, we derive new anisotropic a priori discretization estimates for hexahedral tensor-product meshes, and we provide anisotropic adaptive mesh refinement strategies that are based on a posteriori error estimates, anisotropy indicators, and anisotropic adaptive mesh refinement using hanging nodes.…”

Construction of 4D Simplex Space-Time Meshes for Local Bisection Schemes

“…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.…”