Time to solution I n t e n s i t y R a t e O p e r a t i o n s p e r b y t e t r a n s f e r r e d D Abstract. Important computational physics problems are often large-scale in nature, and it is highly desirable to have robust and high performing computational frameworks that can quickly address these problems. However, it is no trivial task to determine whether a computational framework is performing efficiently or is scalable. The aim of this paper is to present various strategies for better understanding the performance of any parallel computational frameworks for solving PDEs. Important performance issues that negatively impact time-to-solution are discussed, and we propose a performance spectrum analysis that can enhance one's understanding of critical aforementioned performance issues. As proof of concept, we examine commonly used finite element simulation packages and software and apply the performance spectrum to quickly analyze the performance and scalability across various hardware platforms, software implementations, and numerical dis-cretizations. It is shown that the proposed performance spectrum is a versatile performance model that is not only extendable to more complex PDEs such as hydrostatic ice sheet flow equations, but also useful for understanding hardware performance in a massively parallel computing environment. Potential applications and future extensions of this work are also discussed.

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We generate difference and error formulae of arbitrary consistency order on 3 -0 unstructured grids for the finite difference method. Therefore we have to collect grid points so that we can determine the influence polynomials on which the generation of the formulae is based. The problem is to select the appropriate points so that we receive a well-struxtured system of equations of the finite difference method and a good error estimate. We present an aigorithm f o r this selection that i s controlled by two parameters. The high quality of the formulae i s shown b y an example. . References1 SCHONAUER, W.; ADOLPH, T.: How W E solve PDEs, to appear in J. of Computational and Applied Mathematics. 2 SCHMAUDER, M.; ADOLPH, T.: Development of a black-box solver for nonlinear systems of elliptic and parabolic PDEs, to 3 SCHONAUER, W.: Generation of difference and error formulae of arbitrary consistency order on an unstructured grid, in appear in Proc. ENUMATH'99.

The Finite Difference Element Method (FDEM) program package is a robust and efficient black-box solver that solves arbitrary nonlinear systems of elliptic and parabolic partial differential equations under arbitrary nonlinear boundary conditions on arbitrary domains in 2-D and 3-D. FDEM is an unprecedented generalization of the finite difference method on unstructured finite element meshes. From the difference of formulas of different order, we get an easy access to the discretization error. By the knowledge of this error, the mesh may be refined locally to reduce the error to a prescribed relative tolerance. The error estimate is a unique property for such a general black-box. In addition, the FDEM program package is efficiently parallelized on distributed memory parallel computers.In this paper we demonstrate the usefulness of the FDEM program package by its application to several industrial problems. This gives completely new results as up to now people have solved these problems blindly, unaware of the error of their solution. The first problem is the numerical simulation of a microreactor where we have one chemical component entering through the main channel and one chemical component entering through a side channel so that there is a reaction of the components. We want to examine the flow field and the behaviour of the chemical components. The second problem is the heat conduction in a high pressure Diesel injection pump. This problem is based on a preceding fluid-structure interaction problem, and we now compute the temperature distribution in the injection pump additionally. Finally, we simulate the distribution of the temperature in a DC/AC converter module with six power-MOSFETs heated with uniform power. At the bottom of the module air cooling is applied. In contrast to the first two elliptic problems, this is a 3-D parabolic problem. The Finite Difference Element MethodThe main objective of the FDEM program package is the solution of arbitrary nonlinear elliptic and parabolic systems of partial differential equations (PDEs) in 2-D and 3-D with arbitrary nonlinear boundary conditions (BCs) and coupling conditions (CCs). The 3-D operator for the PDEs, BCs and CCs for a system of l equations has the following form: P u ≡ P (t, x, y, z, u, u t , u x , u y , u z , u xx , u xy , u xz , u yy , u yz , u zz ) = 0 .Without t and u t , we have an elliptic problem, otherwise a parabolic one, and without z we get the 2-D operator. For the generation of the difference formulas, we approach the solution locally by an interpolation polynomial P q of order q with m = (q + 1)(q + 2)/2 coefficients in 2-D so that we need m grid points to determine the coefficients. In order to get explicit formulas, we make use of the principle of the influence polynomials that have the value 1 in one node and 0 in the other nodes. The difference formulas are the partial derivatives of the interpolation polynomial.We collect m + r nodes that are sufficient for the order q + ∆q and have to select m appropriate points. The nodes are coll...