Nonlinear Partial Differential Equations - Theory and Numerics

The core activity in this area will still be on analysis of nonlinear PDEs and on numerical methods for PDEs. However, PDEs are central to many of the interdisciplinary projects, and an increasing focus on such research will have the effect that the activity will partly drift in that direction. On the other hand, it is still vital that the group maintains its image as a high-profile research group in the international PDE community. Therefore, a continued effort to pursue activities on more fundamental problems related to PDEs and their numerical discretizations is also necessary.

Many fundamental models in physics and applications have the form of conservation laws. The PDE group has a strong research record in this area, and aims to contribute to the further study of hyperbolic conservation laws and the numerical solutions of these. The initial value problem for conservation laws is not necessarily well posed, and one needs to augment the model with so called entropy
conditions. These depend on the underlying physics, and recently there has been a renewed interest in studying non classical entropy criteria, which result in under-compressive shocks. Such shocks have been observed experimentally in various settings, and we aim to develop further the mathematical framework, which is still in its infancy. Conservation laws with a flux function depending discontinuously on the spatial variables have been studied by members of the PDE group. We aim to continue this study, and to extend previous results, focusing on several spatial dimensions and systems of equations. Furthermore, we also plan to design numerical schemes for such equations and to use these schemes as building blocks for numerical schemes for the MHD equation.

Another class of nonlinear PDEs related to conservation laws is systems of mixed hyperbolic and (degenerate) parabolic equations. An example of such a model is the compressible Navier-Stokes equations. Within this type of model, we aim to make a contribution to the theoretical study of numerical schemes for multidimensional problems. The mathematical understanding of numerical schemes often lags behind their practical implementation. We will study issues related to convergence, and aim to develop methods that are guaranteed to converge. Mixed-type models also occur when modelling solid-liquid separation processes. Therefore numerical methods for such processes are of industrial importance. These models are often very complex, and their mathematical properties are largely unexplored. We plan to study numerical and theoretical problems for these models, with emphasis on convergence and efficiency.

Recently, several members of the PDE group have studied nonlinear variational wave equations. These equations frequently arise as the Euler-Lagrange equations in a variational principle. Among such models are general relativity and waves in liquid crystals. In one spatial dimension, the initial value problem for some of these equations has been partially resolved, but many problems still remain unsolved. Concerning computations, there has been little activity, especially with regard to convergence of numerical schemes, although recently CMA members have proved convergence results for the variational wave, Hunter Saxton, Degasperis-Procesi and the Camassa-Holm equations. We UiO / CMA Winther 146077/V30 Page 8 aim to continue the study of numerical schemes for these types of equations, both theoretically and numerically. Another important issue arising in computations based on conservation laws is that of boundary values. For scalar equations there exists a recent, but well developed, theory regarding well posedness, but for systems of equations this is not so. We aim to study this problem, starting with linear systems, such as the induction equation (which plays an important part in MHD,) and to develop methods for assigning boundary values in practical computations.

The well-posedness of many PDE problems reflects geometrical, algebraic, and topological structures underlying the problem. In recent years, there has been a growing realization that stability of numerical methods, a key property for useful numerical approximations of any PDE problem, can be obtained by constructing discrete schemes which are compatible with these structures. CMA members have contributed to several “break-through” results on for compatible discretization methods for PDEs. This includes the construction by S. Christiansen of new smoothed interpolation operators which commute with the exterior derivative, the development by D. Arnold, R. Falk and R. Winther on the construction of basis functions and degrees of freedom of general piecewise polynomial differential forms of arbitrary order, degree, and dimension, and their use of the Berstein-Gelfand-Gelfand resolution as a design principle for stable mixed finite elements of elasticity. Building on CMA's early leading involvement in this field, the group will pursue a number of follow-up projects which are necessary to utilize the full potential contained in these early papers. Parts of this activity will be related to various improvements of the theory, while there are also a number of possibilities for more applied projects. An example is to develop further the recently proposed elasticity elements into practical computational tools for the challenging problems arising in viscoelasticity and plasticity, where the stress-strain relation is nonlocal. Furthermore, as a part of S. Christiansen's EURYI project, we will have an increased focus on numerical methods for geometric wave equations. There is a need for compatible discretization techniques to overcome parts of difficulties in designing numerical methods for these nonlinear problems, and the newly gained insight on such discretization techniques will benefit this project substantially. The long-term goal of this project is to develop a theoretical foundation on the construction of stable numerical method for geometric wave equations, and as by product, to be able to design effective discretization schemes for the Einstein equation of general relativity in the regime of colliding black holes.

Finally, we will give priority to extended cooperation with CIPR (Centre for Integrated Petroleum Research) at the University of Bergen on mathematical problems arising in the petroleum industry. In particular, we will focus on problems arising from the newly developed industrial technique of using electromagnetic signals to improve the a priori characterizations of oil reservoirs, so called electromagnetic seabed logging. The use of this approach for potential oil reservoirs has been pioneered by the Norwegian oil industry, and the technique is already used world wide as a supplement to the more traditional seismic investigations. However, there is substantial room for improvement in the analysis of the data. The two Norwegian Centres of Excellence, CIPR and CMA, have decided to cooperate on these problems.