We construct monotone numerical schemes for a class of nonlinear PDE for elliptic and initial value problems for parabolic problems. The elliptic part is closely connected to a linear elliptic operator, which we discretize by monotone schemes, and solve the nonlinear problem by iteration. We assume that the elliptic differential operator is in the divergence form, with measurable coefficients s...

We introduce the sparse operator compression to compress a self-adjoint higher-order elliptic operator with rough coefficients and various boundary conditions. The operator compression is achieved by using localized basis functions, which are energy-minimizing functions on local patches. On a regular mesh with mesh size h, the localized basis functions have supports of diameter O(h log(1/h)) an...

Solving elliptic PDEs in more than one dimension can be a computationally expensive task. For some applications characterised by a high degree of anisotropy in the coefficients of the elliptic operator, such that the term with the highest derivative in one direction is much larger than the terms in the remaining directions, the discretized elliptic operator often has a very large condition numb...

In previous papers the arithmetic of hierarchical matrices has been described, which allows to compute the inverse, for instance, of finite element stiffness matrices discretising an elliptic operator L. The required computing time is up to logarithmic factors linear in the dimension of the matrix. In particular, this technique can be used for the computation of the discrete analogue of a resol...

We formulate the elliptic differential operator with infinite number of variables and investigate that it is well defined on infinite tensor product of spaces of square integrable functions. Under suitable conditions, we prove Garding's inequality for this operator.

CONTENTS Introduction. 0. Pseudodifferential operators and linear PDE. §0.1 The Fourier integral representation and symbol classes §0.2 Schwartz kernels of pseudodifferential operators §0.3 Adjoints and products §0.4 Elliptic operators and parametrices §0.5 L 2 estimates §0.6 Gårding's inequality §0.7 The sharp Gårding inequality §0.8 Hyperbolic evolution equations §0.9 Egorov's theorem §0.10 M...

This is arguably one of the deepest and most beautiful results in modern geometry, and in my view is a must know for any geometer/topologist. It has to do with elliptic partial differential operators on a compact manifold, namely those operators P with the property that dim ker P, dim coker P < ∞. In general these integers are very difficult to compute without some very precise information abou...

one can quantize an invertible symbol σ ∈ C∞(S∗X; hom(π∗E, π∗F )) as an elliptic pseudodifferential operator in the Φ-b or edge calculus, ΨΦ-b(X;E,F ), introduced in [7] or alternately as an elliptic operator in the Φ-c or φ calculus, ΨΦ-c(X;E,F ), introduced in [8]. As on a closed manifold, either of these operators will induce a bounded operator acting between natural L-spaces of sections but...