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We propose a novel finite element method scheme for singularly perturbed advection-diffusion-reaction problems, which combines certain quantum-assisted stabilization scheme with a classical h-adaptive approach to provide automatic error…
Calculations of the photonic band structure, transmission coefficients, and quality factors of various two-dimensional, periodic and aperiodic, dielectric photonic crystals by using the finite element method (FEM) are reported. The…
Tumor growth is associated with cell invasion and mass-effect, which are traditionally formulated by mathematical models, namely reaction-diffusion equations and biomechanics. Such models can be personalized based on clinical measurements…
In this paper, we first discuss the optimal convergence of the adaptive finite element methods for non-self-adjoint eigenvalue problems. We present new theoretical error estimators and computable error estimators for multiple and clustered…
The estimation of distributed parameters in partial differential equations (PDE) from measures of the solution of the PDE may lead to under-determination problems. The choice of a parameterization is a usual way of adding a-priori…
The paper studies a finite element method for computing transport and diffusion along evolving surfaces. The method does not require a parametrization of a surface or an extension of a PDE from a surface into a bulk outer domain. The…
We present a novel probabilistic finite element method (FEM) for the solution and uncertainty quantification of elliptic partial differential equations based on random meshes, which we call random mesh FEM (RM-FEM). Our methodology allows…
We introduce the multivariate decomposition finite element method for elliptic PDEs with lognormal diffusion coefficient $a=\exp(Z)$ where $Z$ is a Gaussian random field defined by an infinite series expansion $Z(\boldsymbol{y}) =…
From a systems biology perspective the majority of cancer models, although interesting and providing a qualitative explanation of some problems, have a major disadvantage in that they usually miss a genuine connection with experimental…
A precise domain triangulation is recognized as indispensable for the accurate numerical approximation of differential operators within collocation methods, leading to a substantial reduction in discretization errors. An efficient finite…
In this survey article, a variety of systems modeling tumor growth are discussed. In accordance with the hallmarks of cancer, the described models incorporate the primary characteristics of cancer evolution. Specifically, we focus on…
Objective: Alterations of Young's modulus (YM) and Poisson's ratio (PR) in biological tissues are often early indicators of the onset of pathological conditions. Knowledge of these parameters has been proven to be of great clinical…
The work presents a study of the non-linear mathematical model of tumor growth, proposed by Kolev and Zubik-Kowal (2011). The model is described by a system composed of four partial differential equations that represent the evolution of the…
We present a methodology combining neural networks with physical principle constraints in the form of partial differential equations (PDEs). The approach allows to train neural networks while respecting the PDEs as a strong constraint in…
We consider a two-component reaction-diffusion system that has previously been developed to model invasion of cells into a resident cell population. The system is an idealised version of models of tumour growth in which tumour cells degrade…
This paper provides a unified mathematical analysis of a family of non-local diffuse interface models for tumor growth describing evolutions driven by long-range interactions. These integro-partial differential equations model cell-to-cell…
The present paper deals with the study of the fluence rate over both healthy and tumor tissues in the presence of focal laser ablation (FLA). We propose new analytical solutions for the coupled partial differential equations (PDE) system,…
We present a novel image-based adaptive domain decomposition FEM framework to accelerate the solution of continuum damage mechanics problems. The key idea is to use image-processing techniques in order to identify the moving interface…
In this work, we present an adjoint-based method for discovering the underlying governing partial differential equations (PDEs) given data. The idea is to consider a parameterized PDE in a general form and formulate a PDE-constrained…
In this article, we present a multispecies reaction-advection-diffusion partial differential equation (PDE) coupled with linear elasticity for modeling tumor growth. The model aims to capture the phenomenological features of glioblastoma…