Christian Waluga

Simon Bauer, Daniel Drzisga, Marcus Mohr et al.

We present a novel approach to fast on-the-fly low order finite element assembly for scalar elliptic partial differential equations of Darcy type with variable coefficients optimized for matrix-free implementations. Our approach introduces a new operator that is obtained by appropriately scaling the reference stiffness matrix from the constant coefficient case. Assuming sufficient regularity, an a priori analysis shows that solutions obtained by this approach are unique and have asymptotically optimal order convergence in the $H^1$- and the $L^2$-norm on hierarchical hybrid grids. For the pre-asymptotic regime, we present a local modification that guarantees uniform ellipticity of the operator. Cost considerations show that our novel approach requires roughly one third of the floating-point operations compared to a classical finite element assembly scheme employing nodal integration. Our theoretical considerations are illustrated by numerical tests that confirm the expectations with respect to accuracy and run-time. A large scale application with more than a hundred billion ($1.6\cdot10^{11}$) degrees of freedom executed on 14,310 compute cores demonstrates the efficiency of the new scaling approach.

Markus Huber, Lorenz John, Petra Pustejovska et al.

This article proposes modifications to standard low order finite element approximations of the Stokes system with the goal of improving both the approximation quality and the parallel algebraic solution process. Different from standard finite element techniques, we do not modify or enrich the approximation spaces but modify the operator itself to ensure fundamental physical properties such as mass and energy conservation. Special local a~priori correction techniques at re-entrant corners lead to an improved representation of the energy in the discrete system and can suppress the global pollution effect. Local mass conservation can be achieved by an a~posteriori correction to the finite element flux. This avoids artifacts in coupled multi-physics transport problems. Finally, hardware failures in large supercomputers may lead to a loss of data in solution subdomains. Within parallel multigrid, this can be compensated by the accelerated solution of local subproblems. These resilient algorithms will gain importance on future extreme scale computing systems.

Markus Huber, Ulrich Rüde, Christian Waluga et al.

The Stokes system with constant viscosity can be cast into different formulations by exploiting the incompressibility constraint. For instance the strain in the weak formulation can be replaced by the gradient to decouple the velocity components in the different coordinate directions. Thus the discretization of the simplified problem leads to fewer nonzero entries in the stiffness matrix. This is of particular interest in large scale simulations where a reduced memory bandwidth requirement can help to significantly accelerate the computations. In the case of a piecewise constant viscosity, as it typically arises in multi-phase flows, or when the boundary conditions involve traction, the situation is more complex, and one has to treat the cross derivatives in the original Stokes system with care. A naive application of the standard vectorial Laplacian results in a physically incorrect solution, while formulations based on the strain increase the computational effort everywhere, even when the inconsistencies arise only from an incorrect treatment in a small fraction of the computational domain. Here we propose a new approach that is consistent with the strain-based formulation and preserves the decoupling advantages of the gradient-based formulation in isoviscous subdomains. The modification is equivalent to locally changing the discretization stencils, hence the more expensive discretization is restricted to a lower dimensional interface, making the additional computational cost asymptotically negligible. We demonstrate the consistency and convergence properties of the method and show that in a massively parallel setup, the multigrid solution of the resulting discrete systems is faster than for the classical strain-based formulation. Moreover, we give an application example which is inspired by geophysical research.