University of Manitoba
Winnipeg, Manitoba, Canada
The finite element method (FEM) has well developed in theory, algorithms, implementation, software, and applications for more than fifty years, and has now become the most powerful tool of scientific and engineering computing. One of most significant achievements of FEM in last two decades is the establishing and developing of the p and h-p version of FEM. The former achieves the optimal convergence by increasing the degree of polynomials, and the latter by increasing the degree and refine the mesh, simultaneously.
Because of the use of high-order polynomials, the mathematical theory and methodology for analyzing the performance of the p and h-p FEM are substantially different from those for the traditional h-version of FEM which uses low and fixed degrees and reduces the error by refining mesh. The p version and the h-p version with quasi-uniform mesh were addressed first in early 1980's. and it was proved by Babuska, Szabo, Suri, and others that it converges at least as fast as the h-version and converges as twice fast as the h-version if singularities of the solutions occur at the vertices and edges of non-smooth domains. Although it have developed very well since then, the most effective mathematical tool to analyze the approximation error of the p version was found in very later 1990's by Guo and Babuska, which is the theory of weighted Besov spaces with Jacobi weights. In this theoretical frame the optimal rate of convergence, direct and inverse theorems of the p and h-p version with quasi-uniform mesh have been strictly proved by Guo and Babuska very recently. It is expected that new theory and methodology will bring the research and practices of the p version of FEM to a new level in next century to answer remaining difficulties, for instance, the a-posteriori error estimates and adaptive process for the p-version. On the contrary, the h-p version with geometric mesh was guided at the very beginning by the regularity theory in the frame of the countably normed spaces established by Guo and Babuska in the middle of 1980's for two dimensions and in the middle of 1990's for three dimensions. It has been rigorously proved by Guo and Babuska under the guidance of this theory that it converges exponentially even if the severe singularity occurs. These theoretical and computational accomplishments have had fundamental impacts on the finite element method as well as the theory of regularity theory of differential equations.
The systems of linear algebraic equations generated by the discretization of the p and h-p FEM are extremely large, particularly in three dimensions. It is well-known that iterative method in parallel environment is much more effective than direct solvers, and that the effectiveness of the iterative solvers solely depends on the condition number of the systems. Unfortunately the condition numbers for the p and h-p FEM are very poor. How to improve the condition numbers and design parallel and iterative solvers becomes an important issue theoretically and practically. In 1980's Babuska, Mandel and others proved that the condition numbers for properly preconditioned system is of order $O(1+\ln p)^2$, where $p$ denotes the uniform degree of the polynomials, for the p version in two dimension, and W idlund and Pavarino proved the same results for the p version in three dimensions in 1990's. For the general setting of the h-p version with quasi-uniform mesh or non quasi-uniform mesh, it is proved by Guo and Cao that the condition numbers of preconditioned systems are of order $\max\limits_i(1+\ln {{\textstyle H_ip_i}\over{\textstyle h_i}})^2$ in 1995 for two dimensions and in 1996 for three dimensions, where $H_i$ is the diameter of the $i$-th subdomain, $h_i$ and $p_i$ are the diameter of elements and the maximum poynomial degree used in the subdomain. It is significant that this result reduces to the results for the h and p version as special cases of the h-p version. Moreover, the solvers based on domain decomposition can be easily implemented in fully parallel manner on element level and subdomain level. At each iteration the boundary value problem is decomposed into many small-size subproblems which can be solved parallelly and economically. Hence it provides not only an effective parallel and iterative solver for the linear algebraic system, but also an effective decomposition of boundary value problems of differential equations.
The p and h-p FEM is originated from structural mechanics, but today its use has been widely expanded to various fields of computational mechanics, computational engineering and computational sciences such as elasticity and plasticity, fluid mechanics, composite materials, plate and shell, thermal analysis, electric-magnetic field, Navier-Stokes equation. Among the contributors they are Andresson, Babuska, Demkowicz, Guo, Li, Oden, Schwab, Suri, Szabo, and many others. The techniques and methodology of the p and h-p version developed in the past decades have significantly influenced the industry of commercial finite element codes. There have been many large-scale and general-purpose commercial codes in the world, which have full or partial capacity of the p and h-p version, such as MSC/PROBE, FIESTA, MECHANICAL, PHILEX, STRESSCHECK, and STRIP.
This presentation will survey the fundamentals of the p and h-p finite element method, great accomplishments in the twentieth century and new challenging in the coming twenty-first century.